2-satisfiability, Applications, Renamable Horn satisfiability
To do so, Lewis sets up a 2-satisfiability instance with one variable for each variable of the renamable Horn instance, where the 2-satisfiability variables indicate whether or not to negate the corresponding renamable Horn variables. In order to produce a Horn instance, no two variables that appear in the same clause of the renamable Horn instance should appear positively in that clause; this constraint on a pair of variables is a 2-satisfiability constraint. By finding a satisfying assignment to the resulting 2-satisfiability instance, Lewis shows how to turn any renamable Horn instance into a Horn instance in polynomial time. By breaking up long clauses into multiple smaller clauses, and applying a linear-time 2-satisfiability algorithm, it is possible to reduce this to linear time.


2-satisfiability, Applications, Other applications
2-satisfiability has also been applied to problems of recognizing undirected graphs that can be partitioned into an independent set and a small number of complete bipartite subgraphs, inferring business relationships among autonomous subsystems of the internet, and reconstruction of evolutionary trees.


2-satisfiability, Complexity and extensions, NL-completeness
A nondeterministic algorithm for determining whether a 2-satisfiability instance is not satisfiable, using only a logarithmic amount of writable memory, is easy to describe: simply choose (nondeterministically) a variable v and search (nondeterministically) for a chain of implications leading from v to its negation and then back to v. If such a chain is found, the instance cannot be satisfiable. By the Immerman–Szelepcsényi theorem, it is also possible in nondeterministic logspace to verify that a satisfiable 2-satisfiability instance is satisfiable.


2-satisfiability, Complexity and extensions, NL-completeness
2-satisfiability is NL-complete, meaning that it is one of the "hardest" or "most expressive" problems in the complexity class NL of problems solvable nondeterministically in logarithmic space. Completeness here means that a deterministic Turing machine using only logarithmic space can transform any other problem in NL into an equivalent 2-satisfiability problem. Analogously to similar results for the more well-known complexity class NP, this transformation together with the Immerman–Szelepcsényi theorem allow any problem in NL to be represented as a second order logic formula with a single existentially quantified predicate with clauses limited to length 2. Such formulae are known as SO-Krom. Similarly, the implicative normal form can be expressed in first order logic with the addition of an operator for transitive closure.


2-satisfiability, Complexity and extensions, The set of all solutions
The set of all solutions to a 2-satisfiability instance has the structure of a median graph, in which an edge corresponds to the operation of flipping the values of a set of variables that are all constrained to be equal or unequal to each other. In particular, by following edges in this way one can get from any solution to any other solution. Conversely, any median graph can be represented as the set of solutions to a 2-satisfiability instance in this way. The median of any three solutions is formed by setting each variable to the value it holds in the majority of the three solutions. This median always forms another solution to the instance.


2-satisfiability, Complexity and extensions, The set of all solutions
Feder (1994) describes an algorithm for efficiently listing all solutions to a given 2-satisfiability instance, and for solving several related problems. There also exist algorithms for finding two satisfying assignments that have the maximal Hamming distance from each other.


2-satisfiability, Complexity and extensions, Counting the number of satisfying assignments
#2SAT is the problem of counting the number of satisfying assignments to a given 2-CNF formula. This counting problem is #P-complete, which implies that it is not solvable in polynomial time unless P = NP. Moreover, there is no fully polynomial randomized approximation scheme for #2SAT unless NP = RP and this even holds when the input is restricted to monotone 2-CNF formulas, i.e., 2-CNF formulas in which each literal is a positive occurrence of a variable.


2-satisfiability, Complexity and extensions, Counting the number of satisfying assignments
The fastest known algorithm for computing the exact number of satisfying assignments to a 2SAT formula runs in time O(1.2377n){\displaystyle O(1.2377^{n})}.


2-satisfiability, Complexity and extensions, Random 2-satisfiability instances
One can form a 2-satisfiability instance at random, for a given number n of variables and m of clauses, by choosing each clause uniformly at random from the set of all possible two-variable clauses. When m is small relative to n, such an instance will likely be satisfiable, but larger values of m have smaller probabilities of being satisfiable.


2-satisfiability, Complexity and extensions, Random 2-satisfiability instances
More precisely, if m/n is fixed as a constant α ≠ 1, the probability of satisfiability tends to a limit as n goes to infinity: if α < 1, the limit is one, while if α > 1, the limit is zero. Thus, the problem exhibits a phase transition at α = 1.


2-satisfiability, Complexity and extensions, Maximum-2-satisfiability
In the maximum-2-satisfiability problem (MAX-2-SAT), the input is a formula in conjunctive normal form with two literals per clause, and the task is to determine the maximum number of clauses that can be simultaneously satisfied by an assignment. Like the more general maximum satisfiability problem, MAX-2-SAT is NP-hard. The proof is by reduction from 3SAT.


2-satisfiability, Complexity and extensions, Maximum-2-satisfiability
By formulating MAX-2-SAT as a problem of finding a cut (that is, a partition of the vertices into two subsets) maximizing the number of edges that have one endpoint in the first subset and one endpoint in the second, in a graph related to the implication graph, and applying semidefinite programming methods to this cut problem, it is possible to find in polynomial time an approximate solution that satisfies at least 0.940... times the optimal number of clauses. A balanced MAX 2-SAT instance is an instance of MAX 2-SAT where every variable appears positively and negatively with equal weight. For this problem, Austrin has improved the approximation ratio to min{(3−cosθ)−1(2+(2/π)θ):π/2≤θ≤π}=0.943...{\displaystyle \min \left\{(3-\cos \theta )^{-1}(2+(2/\pi )\theta ) \,:\,\pi /2\leq \theta \leq \pi \right\}=0.943...}.


2-satisfiability, Complexity and extensions, Maximum-2-satisfiability
If the unique games conjecture is true, then it is impossible to approximate MAX 2-SAT, balanced or not, with an approximation constant better than 0.943... in polynomial time. Under the weaker assumption that P ≠ NP, the problem is only known to be inapproximable within a constant better than 21/22 = 0.95454...


2-satisfiability, Complexity and extensions, Maximum-2-satisfiability
Various authors have also explored exponential worst-case time bounds for exact solution of MAX-2-SAT instances.


2-satisfiability, Complexity and extensions, Weighted-2-satisfiability
In the weighted 2-satisfiability problem (W2SAT), the input is an n{\displaystyle n}-variable 2SAT instance and an integer k, and the problem is to decide whether there exists a satisfying assignment in which exactly k of the variables are true.


2-satisfiability, Complexity and extensions, Weighted-2-satisfiability
The W2SAT problem includes as a special case the vertex cover problem, of finding a set of k vertices that together touch all the edges of a given undirected graph. For any given instance of the vertex cover problem, one can construct an equivalent W2SAT problem with a variable for each vertex of a graph. Each edge uv of the graph may be represented by a 2SAT clause u ∨ v that can be satisfied only by including either u or v among the true variables of the solution. Then the satisfying instances of the resulting 2SAT formula encode solutions to the vertex cover problem, and there is a satisfying assignment with  k true variables if and only if there is a vertex cover with k vertices. Therefore, like vertex cover, W2SAT is NP-complete.


2-satisfiability, Complexity and extensions, Weighted-2-satisfiability
Moreover, in parameterized complexity W2SAT provides a natural W[1]-complete problem, which implies that W2SAT is not fixed-parameter tractable unless  this holds for all problems in W[1]. That is, it is unlikely that there exists an algorithm for W2SAT whose running time takes the form f(k)·nO(1). Even more strongly, W2SAT cannot be solved in time no(k) unless the exponential time hypothesis fails.


2-satisfiability, Complexity and extensions, Quantified Boolean formulae
As well as finding the first polynomial-time algorithm for 2-satisfiability, Krom (1967) also formulated the problem of evaluating fully quantified Boolean formulae in which the formula being quantified is a 2-CNF formula. The 2-satisfiability problem is the special case of this quantified 2-CNF problem, in which all quantifiers are existential. Krom also developed an effective decision procedure for these formulae. Aspvall, Plass & Tarjan (1979) showed that it can be solved in linear time, by an extension of their technique of strongly connected components and topological ordering.


2-satisfiability, Complexity and extensions, Many-valued logics
The 2-satisfiability problem can also be asked for propositional many-valued logics. The algorithms are not usually linear, and for some logics the problem is even NP-complete. See Hähnle (2001, 2003) for surveys.


2-sec-Butyl-4,5-dihydrothiazole
2-sec-Butyl-4,5-dihydrothiazole (also known as SBT) is a thiazoline compound with the molecular formula C7H13NS. A volatile pheromone found in rodents such as mice and rats, SBT is excreted in the urine and promotes aggression amongst males while inducing synchronized estrus in females.


2-sec-Butyl-4,5-dihydrothiazole, Binding to MUP
Mouse major urinary proteins (MUPs) are responsible for binding to hydrophobic ligands such as the pheromone SBT. SBT binds within MUP-I's barrel-shaped active site, forming a hydrogen bond with a water molecule within the active site, which in turn is stabilized by forming hydrogen bonds with residue Phe56 and another water molecule; this second water molecule also forms hydrogen bonds to residues in the active site, namely Leu58 and Thr39. SBT also forms van der Waals forces with several of MUP-I's residues, including Ala121, Leu123, Leu134, Leu72, Val100, and Phe108.


2-sec-Butyl-4,5-dihydrothiazole, Binding to MUP
When bound, MUP safely carries SBT through the aqueous environment; once the protein-ligand complex is excreted in the urine, MUP helps prevent SBT decomposition and controls the slow release of SBT over a prolonged period of time, resulting in the physiological and behavioral responses of animals who come into contact with the pheromone.


2-sec-Butyl-4,5-dihydrothiazole, Synthesis
SBT can be produced from 3-(2-aminoethanethio)-4-methylhex-2-enenitrile; however, it is also possible to synthesize SBT from ethanolamine and 2-methylbutanoic acid, using Lawesson's reagent and microwave irradiation.


2-sided
In mathematics, specifically in topology of manifolds, a compact codimension-one submanifold F{\displaystyle F} of a manifold M{\displaystyle M} is said to be 2-sided in M{\displaystyle M} when there is an embedding


2-sided
This means, for example that a curve in a surface is 2-sided if it has a tubular neighborhood which is a cartesian product of the curve times an interval.


2-sided, Examples, Surfaces
For curves on surfaces, a curve is 2-sided if and only if it preserves orientation, and 1-sided if and only if it reverses orientation: a tubular neighborhood is then a Möbius strip. This can be determined from the class of the curve in the fundamental group of the surface and the orientation character on the fundamental group, which identifies which curves reverse orientation.


2-sided, Properties
Cutting along a 2-sided manifold can separate a manifold into two pieces – such as cutting along the equator of a sphere or around the sphere on which a connected sum has been done – but need not, such as cutting along a curve on the torus.


2-sided, Properties
Cutting along a (connected) 1-sided manifold does not separate a manifold, as a point that is locally on one side of the manifold can be connected to a point that is locally on the other side (i.e., just across the submanifold) by passing along an orientation-reversing path.


2-sided, Properties
Cutting along a 1-sided manifold may make a non-orientable manifold orientable – such as cutting along an equator of the real projective plane – but may not, such as cutting along a 1-sided curve in a higher genus non-orientable surface,maybe the simplest example of this is seen when one cut a mobius band along its core curve.


2-step (breakdance move)
The 2-step also known as Mini Swipe or Baby Swipe is a footwork sequence in breakdance.


2-step (breakdance move), Description
The 2-step is a fundamental dance move that is often one of the first footwork sequence learned by breakdancers. Many breakdance moves can begin from the 2-step position. This move sets up the direction of movement and builds up momentum when dancing. This move allows the dancer to stay low and in contact with the ground, which places him in an optimal position for performing other dance moves.


2-step (breakdance move), Description
This move is launched from the third step of the basic 6-step and involves the dancer hopping with a crouched leg over the other straightened leg while simultaneously rotating the dancer's body over the swinging leg in a twisting movement similar to a swipe. The move leaves the dancer in a position similar to the fourth step of the 6-step, but facing 180 degrees in the opposite direction. When dancing 2-step is often used as a move by itself or as a transition into other moves such as powermoves, freezes, toprock and much more.


2-step (breakdance move), Step-by-step
As the name implies, there are a total of two steps in this dance move. A standard version begins in a push up position.


2-step (breakdance move), Step-by-step
Once you have these steps down you should combine steps two and three and do the moves simultaneously.


2-step (breakdance move), Variants
The 2-step is a move based on the 6-step. The 2-step can be down on the knuckles of the hand as well as the palm.


2-step garage
2-step garage, or simply 2-step, is a genre of electronic music and a subgenre of UK garage. One of the primary characteristics of the 2-step sound – the term being coined to describe "a general rubric for all kinds of jittery, irregular rhythms that don't conform to garage's traditional four-on-the-floor pulse" – is that the rhythm lacks the kick drum pattern found in many other styles of electronic music with a regular four-on-the-floor beat.


2-step garage, Characteristics
A typical 2-step drum pattern features syncopated kick drums that skips a beat, with shuffled rhythm or the use of triplets applied to other elements of the percussion, that in particular, results in a sound distinctly different than those present in other house or techno music. Although two kicks to a bar are perceived as having less energy than the four on the floor pattern, the listener's interest is maintained with off-beat snare placements with accents in the drumlines, scattered rimshots in addition to woodblocks,  syncopated basslines, and the percussive use of other instruments such as pads or strings.


2-step garage, Characteristics
Instrumentation usually includes keyboards, synthesizers, and drum machines. Other instruments are added to expand the musical palette, like acoustic recordings, which is likely sampled. The primarily synth basslines used in 2-step are similar to those in the style's progenitors such as UK garage, drum and bass, and jungle. In the other hand, influences from funk and soul can also be heard. Vocals in 2-step garage are usually female, and similar to the style prevalent in house music or contemporary R&B. Some 2-step producers also process and cut up elements of a cappella vocals and use it as an element of the track. Much like other genres derived from UK garage, MCs are often featured, particularly in a live context, with a vocal style reminiscent of old school jungle.


2-step garage, Characteristics
Influences from hip hop and drum and bass, particularly the techstep subgenre have also been noted by critics. The fact that the scene had a significantly different atmosphere to those that surrounded precursors with less aggression at live events was also noted by some critics.


2-step garage, History, Early years
2-step rose to prominence as a genre on jungle and garage-based pirate radio stations in London as an evolution of, and perhaps reaction to developments in contemporary genres such as speed garage, with early 2-step shows often airing at "mellow moments in the weekend" such as Saturday morning or Sunday afternoon. DJs would mix UK garage productions with those of American house and US garage producers such as Masters at Work and Todd Edwards, pitching up the imports to around 130bpm to aid beatmatching. DJs favoured the instrumental (or 'dub') versions of these tracks, because it was possible to play these versions faster without the vocal element of the track sounding odd. Soon, UK producers began to emulate the sound of these pitched-up, imported records in their own tracks.


2-step garage, History, Growing popularity
As the popularity of the sound spread, nights dedicated to it began to crop up, especially in London and Southampton. Label owner and dubstep musician Steve Goodman commented on the Hyperdub website on the debut of Forward>>, a highly influential nightclub in 2-step and later derivatives of the "UK hardcore continuum" – a phrase coined by Goodman to sum up the constant evolution in the hardcore/jungle/garage sound, and later adopted by other writers documenting the scene, such as Martin Clark.


2-step garage, History, Growing popularity
Arguably one of the earliest examples of a 2-step track is the 1997 Kelly G remix of "Never Gonna Let You Go" by Tina Moore, which was a No. 7 hit on the UK Singles Chart. Other notable 2-step hits released in 1997 include "Destiny" by Dem 2 and "The Theme" by the Dreem Teem.


2-step garage, History, 1999–2000: Mainstream success
In 1999 and 2000, 2-step reached the peak of the genre's commercial success. Some critics noted that party organizers favoured 2-step events over nights themed around jungle, drum and bass or other musical precursors because the 2-step nights invited a larger female attendance, and a less aggressive crowd. Much like drum and bass before it, 2-step started to garner crossover appeal, with a collaboration between 2-step producers Artful Dodger and R&B vocalist Craig David reaching #2 in the UK Singles Chart in late 1999 with the song "Re-Rewind".


2-step garage, History, 2000–present: Decline
From 2000 onwards, 2-step as a genre experienced a decline in popularity, but the more experimental releases in the genre from artists such as Horsepower Productions, Zed Bias, Wookie and Steve Gurley stripped away much of the R&B influence of the genre. This style took on a number of names including "dark 2-step", "new dark swing", and the more general term, "dark garage".


2-step garage, History, 2000–present: Decline
This style became a major influence on later styles of UK garage influenced music, such as grime, as well as becoming a direct precursor to dubstep, which took the emphasis on bass and the instrumental nature of later 2-step compositions to their logical conclusion. In 2006, this latter, more experimental style experienced a resurgence in interest, due to the release of the Roots of Dubstep compilation on Tempa, and producers wishing to revisit the roots of the dubstep sound.


2-step garage, History, 2000–present: Decline
Canadian singer The Weeknd's 2016 song "Rockin'" makes use of 2-step sounds.


2-succinyl-6-hydroxy-2,4-cyclohexadiene-1-carboxylate synthase
2-succinyl-6-hydroxy-2,4-cyclohexadiene-1-carboxylate synthase, also known as SHCHC synthase, is encoded by the menH gene in E.coli and functions in the synthesis of vitamin K. The specific step in the synthetic pathway that SHCHC synthase catalyzes is the conversion of 5-enolpyruvoyl-6-hydroxy-2-succinylcyclohex-3-ene-1-carboxylate to (1R,6R)-6-hydroxy-2-succinylcyclohexa-2,4-diene-1-carboxylate and pyruvate.


2-succinyl-6-hydroxy-2,4-cyclohexadiene-1-carboxylate synthase, Background
Vitamin K is a fat soluble vitamin known to aid in blood clotting. It is recommended that all newborns receive an injection of vitamin K in order to prevent excessive bleeding of the brain after birth. There are two major forms of vitamin K that occur naturally. Phylloquinone, also known as K1, is synthesized by plants and is the major form of vitamin K in the diet. Menaquinone, K2, includes a range of forms that are synthesized by bacteria in the gut.


2-succinyl-6-hydroxy-2,4-cyclohexadiene-1-carboxylate synthase, Background
Vitamin K is synthesized from the molecule chorismate in a nine step conversion process. SHCHC synthase catalyzes the third step in the process.


2-succinyl-6-hydroxy-2,4-cyclohexadiene-1-carboxylate synthase, Chemistry, Enzyme Structure
The crystal structure of the MenH enzyme in E.coli (SHCHC synthase) exists as a complex of three protein molecules shown in the diagram. SHCHC synthase forms an alpha/beta hydrolase fold with a central set of seven parallel beta sheets surrounded by alpha helixes on both sides. A cap of five alpha helixes serves to enclose the active site. The enzyme exists in an open form until it binds the substrate, when it morphs into a closed form with an active catalytic triad.


2-succinyl-6-hydroxy-2,4-cyclohexadiene-1-carboxylate synthase, Chemistry, Enzyme Structure
Energetic analysis shows that SHCHC synthase has a low energetic burden for catalytic activity. This means the enzyme is more prone to mutation and is one of the most diverse enzymes in the vitamin K synthetic pathway. Only fifteen amino acid residues are absolutely conserved across mutations of the enzyme.


2-succinyl-6-hydroxy-2,4-cyclohexadiene-1-carboxylate synthase, Chemistry, Catalytic Mechanism
The active site contains a catalytic triad of syrine, histine and arginine, which is conserved across all mutants and is proposed to initiate the reaction. The triad residues are located at Ser86, Asp210, and His232. This triad is proposed to catalyze a proton extraction which triggers a transfer of electrons leading to the elimination of pyruvate and formation of SHCHC. Originally, it was proposed that the transition state was stabilized by a nontraditional oxyanion hole. Now a traditional oxyanion hole is favored, but not definitive.


2-succinyl-6-hydroxy-2,4-cyclohexadiene-1-carboxylate synthase, Chemistry, Cofactors and Alternate Reactions
SHCHC synthase is unaffected by traditional cofactors such as divalent metal ions and EDTA. The enzyme is fairly specific and only acts on SEPHCHC and close derivatives.


2-succinyl-6-hydroxy-2,4-cyclohexadiene-1-carboxylate synthase, Controversy
MenH (SHCHC synthase) was previously thought to be a thioesterase involved in hydrolyzing DHNA-CoA in a later step of menaquinone synthesis. In 2008, it was determined that MenH has poor catalytic activity toward palmitoyl-CoA, casting doubt on its role as a thioesterase. Direct analysis confirmed that MenH is unable to hydrolyze DHNA-CoA. In 2009, it was proposed that a dedicated hotdog fold thioesterase would be needed to catalyze the hydrolysis of DHNA-CoA. A protein was identified in 2013 that could fit this role.


2-tert-Butyl-1,1,3,3-tetramethylguanidine
2-tert-Butyl-1,1,3,3-tetramethylguanidine is an organic base, also known as Barton's base. It is named after Nobel Prize-winning British chemist Derek Barton. Barton and his assistants prepared a series of guanidines with steric hindrance in 1982; in this case five alkyl groups: four methyl groups and one tert-butyl group. These bases were some of the strongest known at the time. In 50% water/ethanol, the acidity constant (pKa) of Barton's base is 14. In acetonitrile its pKa is 24.31.


2-tert-Butyl-1,1,3,3-tetramethylguanidine, Synthesis
The base is prepared by the reaction of tert-butylamine with a Vilsmeier salt. The latter is the reaction product of phosgene with N,N,N′,N′-tetramethylurea.


2-tert-Butyl-1,1,3,3-tetramethylguanidine, Applications
Barton's base can be used in many organic reactions, including in alkylations and in the formation of aziridines. It is often a milder alternative to traditional, strong inorganic bases.


2-transitive group
A 2-transitive group is a transitive group used in group theory in which the stabilizer subgroup of every point acts transitively on the remaining points. Equivalently, a group G{\displaystyle G} acts 2-transitively on a set S{\displaystyle S} if it acts transitively on the set of distinct ordered pairs {(x,y)∈S×S: x≠y}{\displaystyle \{(x,y)\in S\times S:x\neq y\}}. That is, assuming (without a real loss of generality) that G{\displaystyle G} acts on the left of S{\displaystyle S}, for each pair of pairs (x,y),(w,z)∈S×S{\displaystyle (x,y),(w,z)\in S\times S} with x≠y{\displaystyle x\neq y} and w≠z{\displaystyle w\neq z}, there exists a g∈G{\displaystyle g\in G} such that g(x,y)=(w,z){\displaystyle g(x,y)=(w,z)}. Equivalently, gx=w{\displaystyle gx=w} and gy=z{\displaystyle gy=z}, since the induced action on the distinct set of pairs is g(x,y)=(gx,gy){\displaystyle g(x,y)=(gx,gy)}.


2-transitive group, Classifications of 2-transitive groups
Every 2-transitive group is a primitive group, but not conversely. Every Zassenhaus group is 2-transitive, but not conversely. The solvable 2-transitive groups were classified by Bertram Huppert and are described in the list of transitive finite linear groups. The insoluble groups were classified by (Hering 1985) using the classification of finite simple groups and are all almost simple groups.


2-valued morphism
In mathematics, a 2-valued morphism is a homomorphism that sends a Boolean algebra B onto the two-element Boolean algebra 2 = {0,1}. It is essentially the same thing as an ultrafilter on B, and, in a different way, also the same things as a maximal ideal of B. 2-valued morphisms have also been proposed as a tool for unifying the language of physics.


2-valued morphism, Physics
If the elements of B are viewed as "propositions about some object", then a 2-valued morphism on B can be interpreted as representing a particular "state of that object", namely the one where the propositions of B which are mapped to 1 are true, and the propositions mapped to 0 are false. Since the morphism conserves the Boolean operators (negation, conjunction, etc. ), the set of true propositions will not be inconsistent but will correspond to a particular maximal conjunction of propositions, denoting the (atomic) state. (The true propositions form an ultrafilter, the false propositions form a maximal ideal, as mentioned above.)


2-valued morphism, Physics
The transition between two states s1 and s2 of B, represented by 2-valued morphisms, can then be represented by an automorphism f from B to B, such that s2 o f = s1.


2-valued morphism, Physics
The possible states of different objects defined in this way can be conceived as representing potential events. The set of events can then be structured in the same way as invariance of causal structure, or local-to-global causal connections or even formal properties of global causal connections.


2-valued morphism, Physics
The morphisms between (non-trivial) objects could be viewed as representing causal connections leading from one event to another one. For example, the morphism f above leads form event s1 to event s2. The sequences or "paths" of morphisms for which there is no inverse morphism, could then be interpreted as defining horismotic or chronological precedence relations. These relations would then determine a temporal order, a topology, and possibly a metric.


2-valued morphism, Physics
According to, "A minimal realization of such a relationally determined space-time structure can be found". In this model there are, however, no explicit distinctions. This is equivalent to a model where each object is characterized by only one distinction: (presence, absence) or (existence, non-existence) of an event. In this manner, "the 'arrows' or the 'structural language' can then be interpreted as morphisms which conserve this unique distinction".


2-valued morphism, Physics
If more than one distinction is considered, however, the model becomes much more complex, and the interpretation of distinction states as events, or morphisms as processes, is much less straightforward.


2. Amateurliga Bayern
The 2. Amateurliga Bayern was a set of eleven regional leagues in Bavaria existing from 1951 to 1963 as the fourth tier of football in the state.


2. Amateurliga Bayern
The leagues were disbanded in 1963, when the German football league system was reorganised with the introduction of the Bundesliga and replaced by the Landesligas.


2. Amateurliga Bayern, Overview
The 2. Amateurligas were introduced in Bavaria in 1951 to replace the previously existing Kreisligas at this level. This step was taken after the introduction of the 2nd Oberliga Süd as the new second division in Southern Germany in 1950 and the renaming of the Landesliga Bayern to Amateurliga Bayern in 1951. The Landesliga had, until 1950, been functioning as one of the regional second divisions below the Oberliga Süd, alongside the Landesliga Nordbaden, the Landesliga Hessen and the Landesliga Württemberg.


2. Amateurliga Bayern, Overview
The 2. Amateurliga, as the fourth tier of the German league system, below the Amateurliga Bayern as the third, was sub-divided into seven regions, along the boundaries of the seven Regierungsbezirke, those regions being:


2. Amateurliga Bayern, Overview
Some of the seven regions of the 2. Amateurliga were in turn sub-divided into regional leagues, resulting in the existence of eleven leagues at this level:


2. Amateurliga Bayern, Overview
Within the four Regierungsbezirke who had two leagues, a championship was played to determine the local champions, but no overall 2. Amateurliga championship was held. Instead, the regional champions would play promotion rounds to determine the teams that would move up to the Amateurliga. The promotion rounds where sub-divided into northern and southern Bavaria. The champions of the three Franconian Regierungsbezirke would play in the northern group while the other four Regierungsbezirke would play in the southern one. Swabia had a special status within this system, being allowed to send both champions and runners-up to the promotion round. Originally, until 1957, the promotion rounds were held in two groups of six teams, playing a home-and-away round, with the top-two teams from each being promoted.


2. Amateurliga Bayern, Overview
In 1953, the Amateurliga Bayern was split in itself in two regional divisions, north and south, resulting in two new leagues:


2. Amateurliga Bayern, Overview
This decision was made rather late, on 7 July 1953, after the promotion rounds had already been played, resulting in all twelve teams that had taken part being promoted.


2. Amateurliga Bayern, Overview
The promotion system remained unchanged after this, two groups of six with the first two in each group still promoted. In 1955, three teams were promoted from the north because VfB Helmbrechts had been promoted to the 2nd Oberliga Süd, thereby providing an additional spot in the Amateurliga Nordbayern. An expansion of the Amateurliga Südbayern from 14 to 16 clubs in 1957 allowed four teams to be promoted from the south.


2. Amateurliga Bayern, Overview
From 1957-58 onwards, the promotion system was changed. In the north, the three Regierungsbezirk champions were now directly promoted. In the south, the Oberbayern and Schwaben champions were directly promoted while the champions of Niederbayern and Oberpfalz played a decider for one more promotion spot.


2. Amateurliga Bayern, Overview
In 1958, an additional spot was available because the 1. FC Bamberg had been promoted to the second division, this was played out between the three runners-up. The same happened in 1959, when SpVgg Bayreuth moved up to the second division. In 1960, when Schwaben Augsburg was promoted to the 2nd Oberliga, the losing team of the Niederbayern versus Oberpfalz encounter was also promoted.


2. Amateurliga Bayern, Disbanding of the 2. Amateurligas
In 1963, the German league system was changed drastically. The Fußball-Bundesliga replaced the previously existing five Oberligas as the highest league in the country. Below it, five Regionalligas were created as the new second divisions.


2. Amateurliga Bayern, Disbanding of the 2. Amateurligas
On 27 April 1963, it was decided to reorganise the Bavarian league system, too. The two Amateurligas in Bavaria were reduced to a single division again, like until 1953. The 2nd Amateurligas were completely disbanded and replaced by three Landesligas as the new fourth tier, those being:


2. Amateurliga Bayern, Disbanding of the 2. Amateurligas
Apart from the ten clubs each that came from the two Amateurligas to the new Landesligas, the top-teams of the 2. Amateurligas in 1962-63 were also allowed to join the new leagues:


2. Amateurliga Bayern, Disbanding of the 2. Amateurligas
All other clubs from the former 2. Amateurligas went to the Bezirksligas, which had existed since 1957, the fifth tier of league football.


2. Amateurliga Bayern, League champions
The champions of the eleven 2. Amateurligas in Bavaria were:


2. Basketball Bundesliga
The 2. Basketball Bundesliga is the second tier level league of professional club basketball in Germany. Before the 2007–08 season, the 2. Basketball Bundesliga consisted of two geographical divisions (North and South). Since 2007 the 2. Basketball Bundesliga now consists of the two hierarchical leagues, named ProA and ProB. At the end of the season, the top two teams of the ProA qualify for the Basketball Bundesliga, and the teams positioned 15th and 16th are relegated to the lower league ProB.


2. Bundesliga
The 2. Bundesliga (Zweite Bundesliga [ˈtsvaɪtə ˈbʊndəsˌliːɡa]) is the second division of professional football in Germany. It was implemented 11 years after the founding of the Fußball-Bundesliga as the new second division for professional football. The 2. Bundesliga is ranked below the Bundesliga and above the 3. Liga in the German football league system. All of the 2. Bundesliga clubs qualify for the DFB-Pokal, the annual German Cup competition. A total of 127 clubs have competed in the 2. Bundesliga since its foundation.


2. Bundesliga
The decision to establish the league as the second level of football in West Germany was made in May 1973. The league started operating in August 1974, then with two divisions of 20 clubs. It was reduced to a single division in 1981. From the 1991–92 season onwards clubs from former East Germany started participating in the league, briefly expanding it to two divisions again. It returned to a single division format again at the end of that season and has had 18 clubs as its strength since 1994.


2. Bundesliga
Two clubs from the 2. Bundesliga are directly promoted to the Bundesliga, while a third promoted club is determined through play-offs, from 1974 to 1991 and again since 2008. Between 1991 and 2008 the third-placed club in the league was directly promoted. The bottom clubs in the league are relegated to the third division; from 1974 to 1994 the Oberliga, from 1994 to 2008 the Regionalliga and since 2008 the 3. Liga. The number of relegated clubs has fluctuated over the years. Since 2008 two clubs are directly relegated while the third-last team has the opportunity to defend its league place in play-offs against the third placed team of the 3. Liga.


2. Bundesliga
1. FC Nürnberg, SC Freiburg, 1. FC Köln, Arminia Bielefeld and VfL Bochum hold the record number of championships in the league with four each. Bielefeld also holds the record for number of promotions from the 2. Bundesliga to the Bundesliga, with eight.


2. Bundesliga
For the 2018–19 season, an average of 19,128 spectators watched 2. Bundesliga matches, making the 2. Bundesliga the world's second most-watched secondary football league, after the EFL Championship in England.


2. Bundesliga, History, Background
With the implementation of the Bundesliga in 1963, five Regionalligen were also founded as the 2nd highest playing level, South, Southwest, West, North and Berlin. The two top ranking teams from each Regionalliga at the end of a season were placed into groups to play against the relegation teams from the Bundesliga. With the transition from the former Oberliga to the newly created Bundesliga and Regionalliga, however, it became clear that the substructure of the Bundesliga was both sportingly and economically problematic and that relegation from the Bundesliga could easily ruin a club economically.


2. Bundesliga, History, Background
This situation was partly responsible for the Bundesliga scandal in 1971, in which, due to manipulations in point games in the relegation battle, Rot-Weiß Oberhausen and Arminia Bielefeld managed to remain in the Bundesliga. As a consequence of the scandal, the DFB-Bundestag in Frankfurt decided on 30 June 1973 to introduce a 2. Bundesliga, divided into a north and a south season, for the 1974/75 season, which should close the gap between professional and amateur areas. According to an elaborate point system, the clubs should be able to qualify for the new divisions.


2. Bundesliga, History, Background
Not only the placements of the previous five seasons were decisive, but also economic and structural requirements. There was also a five-year evaluation, with the first two years being single, the next two years double and the last year triple. In the event of a tie, the last year should apply.


2. Bundesliga, History, Background
With this regulation, however, the DFB had overlooked that the ten regional leagues participating in the annual promotion round to the Bundesliga had not qualified for the new 2. Bundesliga from the outset due to their placement in the 1973/74 season. In the case of 1. FC Saarbrücken, this meant that the club was accepted into the 2 Bundesliga South after failing in the promotion round and took the place of SV Alsenborn, which also came from the Regionalliga Südwest.


2. Bundesliga, History, Background
SV Alsenborn, a "village club" sponsored by Fritz Walter, had failed three times in the Bundesliga promotion round in 1968, 1969 and 1970 and had clearly qualified for the new division.


2. Bundesliga, History, Background
The DFB found that there were no conditions suitable for the second division in Alsenborn and that they could not be achieved in the foreseeable future and downgraded SV Alsenborn to the then third-class amateur league Southwest, although the club had a successful participation in one of the three rounds of promotion to the Bundesliga Special permission would have been allowed to play even in the top division (of course not in Alsenborn, but in Kaiserslautern or Ludwigshafen).


2. Bundesliga, History, Formation
The decision to establish the 2. Bundesliga as a fully professional league below the Bundesliga was made at the annual convention of the German Football Association, the DFB, in Frankfurt on 30 June 1973. The league replaced the five Regionalligas that were at this level from 1963 to 1974. Each Regionalliga had a set quota of clubs that could qualify for the new league with the Regionalliga Süd receiving thirteen spots, the Regionalliga West twelve, the Regionalliga Nord and Regionalliga Südwest seven and the Regionalliga Berlin one. The qualified teams were established through a ranking that took the last five seasons of the Regionalliga into account.


2. Bundesliga, History, Formation
The new 2. Bundesliga was split into a northern and a southern division with 20 clubs each. Each division had its champion directly promoted to the Bundesliga while the two runners-up would contest a two-leg play-off to determine the third promoted team. The bottom four clubs in each league were relegated, however, as the number of clubs relegated from the Bundesliga to each division could vary, so could the number of clubs in the league and therefore the number of teams relegated.


2. Bundesliga, History, 2. Bundesliga North and South 1974 to 1981
The first-ever game of the league was played on Friday, 2 August 1974 between 1. FC Saarbrücken and Darmstadt 98 and ended in a 1–0 win for Saarbrücken, with Nikolaus Semlitsch scoring the first-ever goal of the new league in the 18th minute of the game. The inaugural champions of the league were Hannover 96 in the north and Karlsruher SC in the south, both former Bundesliga clubs. The play-offs for the third Bundesliga spot were contested by FK Pirmasens and Bayer Uerdingen, with Uerdingen winning 6–0 at home after a four-all draw in the first leg. The three promoted teams however proved uncompetitive in the Bundesliga with Hannover and Uerdingen being relegated straight away again while Karlsruhe lasted for only two seasons.


2. Bundesliga, History, 2. Bundesliga North and South 1974 to 1981
The second season saw league championships for Tennis Borussia Berlin and 1. FC Saarbrücken, with Tennis Borussia lasting for only one season and 1. FCS for two. The contest for the third promotion spot pitted two far bigger names of German football against each other, with Borussia Dortmund edging out 1. FC Nürnberg with two wins, ending Dortmund's four-year second division spell. The last round of the season in the south also saw an all-time goal scoring record per round when 55 goals were scored in ten games. The northern division incidentally set the second best mark when it scored 51 goals the day before.


2. Bundesliga, History, 2. Bundesliga North and South 1974 to 1981
In 1976–77 the league champions were FC St. Pauli and VfB Stuttgart while the third promotion spot went to 1860 Munich, having had to play a third game after Arminia Bielefeld and TSV 1860 each won their home games 4–0, with the decider ending 2–0 in favour of the southern team. Ottmar Hitzfeld set an all-time 2. Bundesliga record in May 1977 when he scored 6 goals in a league match for VfB Stuttgart against Jahn Regensburg.


2. Bundesliga, History, 2. Bundesliga North and South 1974 to 1981
Bielefeld won promotion as the champions of the northern division in the following season, as did southern champion SV Darmstadt 98, entering the Bundesliga for the first time in its history. Third place went to 1. FC Nürnberg who overcame Rot-Weiss Essen with a 1–0 home win and a two–all draw away. For Nürnberg it ended a nine-year absence from the Bundesliga. Horst Hrubesch set an all-time record that season for goals in one season, 41 scored for Rot-Weiss Essen.


2. Bundesliga, History, 2. Bundesliga North and South 1974 to 1981
In 1978–79 direct promotion went to 1860 Munich and Bayer Leverkusen while the play-off was won, once more, by Bayer Uerdingen, which defeated SpVgg Bayreuth 2–1 at home after a draw away. In the north, two clubs were relegated from the league for financial reasons, Westfalia Herne, which had finished fifth and former Bundesliga side FC St. Pauli, which had come sixth. The following seasons saw 1. FC Nürnberg and Arminia Bielefeld clinch another promotion from the 2. Bundesliga, as did Karlsuher SC which overcame Rot-Weiss Essen by winning 5–1 at home after losing 3–1 away. Arminia Bielefeld set an all-time 2. Bundesliga record when it defeated Arminia Hannover 11–0 in May 1980, the biggest-ever win in the league.


2. Bundesliga, History, 2. Bundesliga North and South 1974 to 1981
The 1980–81 season, the seventh of the league, was also its last in this format. From 1981 it played as a single division of 20 teams after a decision taken on 7 June 1980, when, at a special convention of the DFB, the introduction of the single division 2. Bundesliga was decided upon with a majority of 84 votes to 77. The northern division was unusually strong that season, having received all three relegated teams of the 1979–80 Bundesliga season, SV Werder Bremen, Eintracht Braunschweig and Hertha BSC, and playing with 22 teams.


2. Bundesliga, History, 2. Bundesliga North and South 1974 to 1981
Bremen won the league while Braunschweig came second. Hertha missed out despite scoring 123 goals. In the south, the league was won by SV Darmstadt 98 for a second time while runners-up Kickers Offenbach lost out to Braunschweig in the play-offs. The reduction of the league to a single division meant 22 teams were relegated while no team was promoted to the 2. Bundesliga that season.


2. Bundesliga, History, Single division era 1981 to 1991
The new single division league of 20 teams saw only a small change in modus. The top two in the league were promoted while the third placed team played the sixteenth placed Bundesliga side in a home-and-away play-off for one more spot in the Bundesliga. The bottom four in the league were relegated. The inaugural season saw FC Schalke 04 compete in the 2. Bundesliga for the first time, and win it. Second place went to Hertha BSC while third placed Kickers Offenbach missed out on promotion after losing both play-off games to Bayer Leverkusen.


2. Bundesliga, History, Single division era 1981 to 1991
Fourth place went to 1860 Munich, one point behind Offenbach, but the club found itself relegated after the DFB refused it a license for the following season. This decision kept 17th placed SG Wattenscheid 09, the best-placed team on a relegation rank, in the league. The following season finally saw Kickers Offenbach win promotion from the 2. Bundesliga, behind champions SV Waldhof Mannheim who had never played in the Bundesliga before. Bayer Uerdingen, in third place, won promotion through the play-offs for a third time, this time overcoming the previous seasons 2. Bundesliga champions FC Schalke 04.


2. Bundesliga, History, Single division era 1981 to 1991
Schalke bounced back immediately, coming second behind Karlsruher SC in 1983–84. Third place went to MSV Duisburg wo were decisively beaten 0–5 by Bundesliga side Eintracht Frankfurt at home. At the other end, Rot-Weiss Essen, after having failed to win promotion to the Bundesliga through the play-offs twice from the 2. Bundesliga, was relegated to amateur football that season. 1. FC Nürnberg took out the championship of the single division 2. Bundesliga for the first time in 1985, with Hannover 96 coming second.


2. Bundesliga, History, Single division era 1981 to 1991
Third placed 1. FC Saarbrücken also won promotion courtesy to a 2–0 home win over Arminia Bielefeld after a draw away. Kickers Offenbach, freshly relegated from the Bundesliga came only 19th in the 2. Bundesliga, suffered another relegation, as did another former Bundesliga side, FC St. Pauli, having returned to the league for the first time after having had its license revoked in 1979.


2. Bundesliga, History, Single division era 1981 to 1991
In 1985–86, three clubs from Berlin competed in the league, but none the following season, with Blau-Weiß 90 Berlin achieving its one and only promotion to the Bundesliga while Hertha BSC and Tennis Borussia were relegated to amateur football. The league champions were FC 08 Homburg, also promoted to the Bundesliga for the first time. Arguably one of the most famous play-off contests in 2. Bundesliga history however were the games between third placed Fortuna Köln and Borussia Dortmund.


2. Bundesliga, History, Single division era 1981 to 1991
Köln won 2–0 at home, followed by a 3–1 for Dortmund, making a third game necessary as the away goal rule did not apply to the Bundesliga versus 2. Bundesliga play-offs at the time. This third game was won 8–0 by Borussia Dortmund in front of 50,000 in neutral Düsseldorf. In the relegation zone MSV Duisburg followed the two Berlin clubs into amateur football as a third former Bundesliga side that season.


2. Bundesliga, History, Single division era 1981 to 1991
Hannover 96 and Karlsruher SC won promotion once more in 1987 while third placed FC St. Pauli, freshly promoted from amateur football again, missed out by a goal in the play-offs against FC Homburg. At the bottom end Eintracht Braunschweig became another former Bundesliga side and champion to drop into the third division. FC St. Pauli ended a ten-year wait for Bundesliga return in 1988 when it finished runners-up to Stuttgarter Kickers who were promoted to the Bundesliga for the first time.


2. Bundesliga, History, Single division era 1981 to 1991
Third placed SV Darmstadt 98 missed out on penalties in the deciding third game against SV Waldhof Mannheim after each side had won their home game by a goal. Arminia Bielefeld came a distant last and was relegated while 17th placed SpVgg Bayreuth was rescued when Rot-Weiß Oberhausen was refused a license.


2. Bundesliga, History, Single division era 1981 to 1991
Fortuna Düsseldorf won the league championship in 1988–89, with two clubs from the Saarland coming second and third, FC Homburg and 1. FC Saarbrücken. Saarbrücken however was unable to overcome Eintracht Frankfurt in the later club's second successful play-off defence of its Bundesliga place. SpVgg Bayreuth finished 17th again but was again spared from relegation when Kickers Offenbach was refused a license. Also relegated were Union Solingen after 14 consecutive seasons in the league.


2. Bundesliga, History, Single division era 1981 to 1991
At the end of the season Spanish-born Joaquín Montañés retired from 2. Bundesliga football after 479 games for Alemannia Aachen in the league from 1974 to 1989, a record for any player with a single club in the league. In 1990 Hertha BSC completed its return from amateur football to the Bundesliga with a 2. Bundesliga title, followed up by SG Wattenscheid 09, who entered the Bundesliga for the first time.


2. Bundesliga, History, Single division era 1981 to 1991
1. FC Saarbrücken failed in the play-offs for a second consecutive time when it missed out to VfL Bochum, thereby ensuring a Bochum derby in the Bundesliga between VfL and Wattenscheid for the following season. In the relegation zone SpVgg Bayreuth failed to get reprieved for a third consecutive season and dropped into amateur football, as did Alemannia Aachen, a founding member of the 2. Bundesliga who had played all 16 seasons of the league until then.


2. Bundesliga, History, Single division era 1981 to 1991
The tenth season of the single division 2. Bundesliga was to be the last in its current format for a time as the German reunification in 1991 lead to changes to the league after this season. With FC Schalke 04 and MSV Duisburg two long-term Bundesliga teams finished at the top of the league. In third place Stuttgarter Kickers had to play FC St. Pauli three times to earn promotion, the first two contests having ended 1–1 while Stuttgart won the third 3–1. FC Schweinfurt 05 in last place became one of the worst clubs in the league history when it only won two games all season. Rot -Weiss Essen had its license revoked which allowed SV Darmstadt 98 to avoid relegation.


2. Bundesliga, History, German reunification 1991–92
In the 1991–92 season, the league was expanded to 24 teams in two regional divisions, north and south, to accommodate six new East German clubs which joined the league that season. The East German clubs were spread very unevenly, with one going to the north and five to the south, caused by the geographic location of those clubs. Only the league champions were promoted to the Bundesliga that year, which were Bayer Uerdingen in the north and 1. FC Saarbrücken in the south. The bottom three in each division were relegated, three of which were from former East Germany.


2. Bundesliga, History, German reunification 1991–92
The other two were former Bundesliga clubs, Blau-Weiß 90 Berlin and 1860 Munich, with the later having played its first season back in the 2. Bundesliga after their license was revoked in 1982. At the end of this season the league returned to the single division format, but with still 24 clubs as its strength.


2. Bundesliga, History, Single division era 1992 to present
The 1992–93 season was a momentous one, with 24 teams competing in a single league and each club playing 46 games. Three clubs were promoted directly, as would be the case from then on until 2008, with the play-offs having been abolished. SC Freiburg won the league and promotion for the first time. Behind it, MSV Duisburg made a return to the Bundesliga while third placed VfB Leipzig became the first former East German club to earn promotion from the 2. Bundesliga.


2. Bundesliga, History, Single division era 1992 to present
Seven clubs were relegated from the league to reduce its strength to 20 clubs again from the following season. Of those Eintracht Braunschweig, Fortuna Düsseldorf and SV Darmstadt 98 were former Bundesliga sides. The following season saw changes again as it was the last with 20 clubs. Promoted were VfL Bochum, Bayer Uerdingen and 1860 Munich, which had just won promotion from the third division the year before and returned to the Bundesliga for the first time since 1981. At the bottom end, five clubs were relegated, four of those former Bundesliga sides and the fifth one, Carl Zeiss Jena, from former East Germany.


2. Bundesliga, History, Single division era 1992 to present
The league level below the 2. Bundesliga was changed profoundly in 1994 with the Oberligas replaced by the new Regionalligas, which allowed for direct promotion to the 2. Bundesliga for the first time since 1980. The league itself was now reduced to 18 clubs with no play-offs, three promoted and four relegated teams, a system it would maintain until 2008, when the play-offs were re-introduced. Hansa Rostock won the 2. Bundesliga for the first time in 1995 and FC St. Pauli and Fortuna Düsseldorf followed it up to the Bundesliga. In the relegation zone FSV Frankfurt came a distant last with only three wins to its name while the two Saarland sides FC Homburg and 1. FC Saarbrücken accompanied it. The later, despite finishing seventh, had its license revoked, thereby sparing FSV Zwickau from relegation.


2. Bundesliga, History, Single division era 1992 to present
The 1995–96 season saw VfL Bochum win the league again with second placed Arminia Bielefeld winning promotion straight after having been promoted from the Regionalliga the year before. Third place went to MSV Duisburg while Hannover 96, 1. FC Nürnberg and SG Wattenscheid 09 were all former Bundesliga clubs now suffering relegation to the third division. The 1. FC Kaiserslautern and Eintracht Frankfurt had suffered their first-ever relegation from the Bundesliga in 1996. The former won the league and bounced back immediately while Frankfurt remained at this level for another season.


2. Bundesliga, History, Single division era 1992 to present
Kaiseslautern was accompanied up by VfL Wolfsburg, who won promotion for the first time, and Hertha BSC. Kaiserslautern would also become the first and only club to win the Bundesliga as a freshly promoted side the following year. The 1. FC Kaiserslautern and SV Meppen also set a record for number of goals in a game, 13, when Kaiserslautern defeated Meppen 7–6. Eintracht Frankfurt won the league in 1998 with SC Freiburg coming second while 1. FC Nürnberg, freshly returned from the Regionalliga, came third. At the bottom end VfB Leipzig was one of three clubs from the east to be relegated, alongside SV Meppen, which dropped out of the league after eleven consecutive seasons there.


2. Bundesliga, History, Single division era 1992 to present
The 1998–99 season saw the 1. FC Köln in the league for the first time, having been relegated from the Bundesliga after 35 consecutive seasons there from the start of the league in 1963. Köln only managed to come tenth, while the league was won by Arminia Bielefeld. Behind Arminia Bielefeld, SpVgg Unterhaching and SSV Ulm 1846 entered the Bundesliga for the very first time. Last place in the league went to Fortuna Düsseldorf, which was accompanied to the Regionalliga by SG Wattenscheid 09, KFC Uerdingen 05, formerly Bayer Uerdingen, and FC Gütersloh.


2. Bundesliga, History, Single division era 1992 to present
The first season of the new millennium saw the end of an era, with Fortuna Köln being relegated after 26 consecutive seasons in the league since the start in 1974. Local rival 1. FC Köln won the league while VfL Bochum came second and FC Energie Cottbus, in third place, moved up to the Bundesliga for the first time. Fortuna Köln was accompanied to the Regionalliga by Karlsruher SC, Kickers Offenbach and Tennis Borussia Berlin, who had their license revoked.


2. Bundesliga, History, Single division era 1992 to present
In 2000–01, the league was won by 1. FC Nürnberg once again, with Borussia Mönchengladbach earning promotion back to the Bundesliga after a two-year absence. FC St. Pauli was the third promoted team. SSV Ulm 1846, freshly relegated from the Bundesliga, finished the season in 16th place and became insolvent. Hannover 96, Arminia Bielefeld and VfL Bochum were the promoted teams in 2002, while the following season saw 1. FC Köln and Eintracht Frankfurt competing and succeeding for promotion again, behind league champions SC Freiburg.


2. Bundesliga, History, Single division era 1992 to present
In 2004, 1. FC Nürnberg and Arminia Bielefeld earned another one of their many promotions while third placed 1. FSV Mainz 05 was a newcomer to the Bundesliga. Like in 2003, 2005 saw 1. FC Köln and Eintracht Frankfurt win promotion while between them, in second place, MSV Duisburg moved up, too. At the bottom end three of the four relegated clubs shared similar names, Rot-Weiß Oberhausen, Rot-Weiss Essen and Rot-Weiß Erfurt with the fourth team relegated being Eintracht Trier.


2. Bundesliga, History, Single division era 1992 to present
VfL Bochum won the league again in 2006 while FC Energie Cottbus returned to the Bundesliga for a second three-year stint. In second place Alemannia Aachen returned to the Bundesliga for the first time since 1970. Relegated that year were Dynamo Dresden, former East German power house, after a two-year stint in the league. 2006 also saw the retirement of Willi Landgraf from 2. Bundesliga football. Landgraf had played a record 508 2. Bundesliga games from 1986 to 2006, playing in the league for Rot-Weiss Essen, FC 08 Homburg, FC Gütersloh and Alemannia Aachen.


2. Bundesliga, History, Single division era 1992 to present
Karlsruher SC ended an absence from the Bundesliga that had lasted since 1998 when it won the league in 2007 and was followed up by Hansa Rostock and MSV Duisburg. Freshly relegated Borussia Mönchengladbach won the league the following year, with new Bundesliga club TSG 1899 Hoffenheim second and 1. FC Köln third.


2. Bundesliga, History, Single division era 1992 to present
The 2008–09 season saw the return of play-offs. The third placed team in the 2. Bundesliga now played the 16th placed team in the Bundesliga for a spot in that league. At the other end of the table, the 16th placed 2. Bundesliga side would now also play the third placed team in the new 3. Liga, which had replaced the Regionalliga as the third division. SC Freiburg and 1. FSV Mainz 05 were directly promoted that season while 1. FC Nürnberg had to enter the play-offs in which it defeated FC Energie Cottbus 5–0 on aggregate. At the relegation end, VfL Osnabrück lost its 2. Bundesliga place to SC Paderborn from the 3. Liga.


2. Bundesliga, History, Single division era 1992 to present
1. FC Kaiserslautern ended a four-year spell in the 2. Bundesliga in 2010 with a league championship, with FC St. Pauli coming second. The FC Augsburg finished third but was unable to overcome 1. FC Nürnberg in the play-offs and lost 3–0 on aggregate. Hansa Rostock, in 16th place, dropped out of the 2. Bundesliga when it lost both play-off games to Ingolstadt 04. Hertha BSC and FC Augsburg were directly promoted to the Bundesliga in 2010, the later for the first time, while VfL Bochum in third place missed out on promotion against Borussia Mönchengladbach. VfL Osnabrück found itself unsuccessfully defending its league place again, losing to Dynamo Dresden in extra time in the second leg.


2. Bundesliga, History, Single division era 1992 to present
After 15 consecutive seasons in the 2. Bundesliga a numerous attempts at promotion Greuther Fürth finally won the league in 2012. Eintracht Frankfurt came second and Fortuna Düsseldorf returned to the Bundesliga for the first time since 1997 when it defeated Hertha BSC in the play-offs. Karlsruher SC failed to remain in the 2. Bundesliga when it was relegated on away goal rule after two drawn games against Jahn Regensburg.


2. Bundesliga, History, Single division era 1992 to present
Hertha BSC won the 2. Bundesliga for the second time in three seasons in 2012–13 and was accompanied up by Eintracht Braunschweig, who had not played in the Bundesliga since 1985. Third placed 1. FC Kaiserslautern lost both games to 1899 Hoffenheim and thereby failed to get promoted. Dynamo Dresden became the first 2. Bundesliga side in five attempts to hold onto their league place while 3. Liga side VfL Osnabrück missed out in the play-offs for a third time in three attempts.


2. Bundesliga, History, Single division era 1992 to present
The 2013–14 season ended with 1. FC Köln winning the league, followed up by SC Paderborn who won promotion to the Bundesliga for the first time. Relegated where Energie Cottbus and Dynamo Dresden, both former Bundesliga sides. Third placed SpVgg Greuther Fürth failed to gain promotion after two draws with Bundesliga club Hamburger SV. At the bottom end two eastern clubs were relegated, Dynamo Dresden and Energie Cottbus, while Arminia Bielefeld entered the relegation round.


2. Bundesliga, History, Single division era 1992 to present
The 2014–15 season saw Ingolstadt 04 win the league and earn Bundesliga promotion for the first time while SV Darmstadt 98 finished second and returned to the Bundesliga for the first time since 1982. Third placed Karlsruher SC faced Hamburg for another Bundesliga spot while TSV 1860 Munich had to play Holstein Kiel to retain their place in the 2. Bundesliga. Both the Bundesliga and 2. Bundesliga clubs retained their league membership. The two clubs directly relegated from the league where Erzgebirge Aue and VfR Aalen.


2. Bundesliga, History, Single division era 1992 to present
In the 2015–16 the league was won by SC Freiburg, with RB Leipzig finishing runners-up and earning its first-ever Bundesliga promotion, while 1. FC Nürnberg had to enter the promotion play-off where it lost on aggregate to Eintracht Frankfurt. At the bottom end of the table SC Paderborn suffered consecutive relegations, becoming the sixth club to drop from the Bundesliga to the third tier in consecutive seasons. FSV Frankfurt was the second team directly relegated while MSV Duisburg entered the relegation play-off where it was unsuccessful. All the clubs promoted to the 2. Bundesliga had played there before but while Dynamo Dresden and Erzgebirge Aue had only experienced a short absence the third club, Würzburger Kickers, had not played at this level for almost 40 years.


2. Bundesliga, History, Single division era 1992 to present
In December 2016, it was announced that for the first time, the 2. Bundesliga would be given its own logo, taking effect at the start of the 2017–18 season. Previously, the 2. Bundesliga had borrowed the Bundesliga logo for promotional and media purposes. The 2. Bundesliga gained its own logo to "strengthen the profile of the competition" and to better identify the league with fans, the media, and sponsors.


2. Bundesliga, History, Single division era 1992 to present
After their relegation, the 2018–19 season was Hamburger SV's first season outside of the German top flight in their 55-year history.


2. Bundesliga, History, All-time table
The most consistent team in the league, as of 2020–21, is SpVgg Greuther Fürth, having won 1,667 points from 1,126 games. In second place sits FC St. Pauli, best placed of the current 2. Bundesliga clubs. Alemannia Aachen comes third in the all-time table, while last place, number 127, goes to Spandauer SV with just ten points to its name.


2. Bundesliga, Members of the 2. Bundesliga (2021–22 season)
For details on the 2. Bundesliga 2021–22 season, see here.


2. Bundesliga, League rules
Since the 2006–07 season there is no longer a limit on non-EU players in the league. Instead clubs are required to have 8 players on the squad who have come up through the youth system of a German club, 4 of which have to come from the club's own youth system. Seven substitutes are permitted to be selected, from which three can be used in the duration of the game.


2. Bundesliga, Promoted and relegated teams
The list of teams that earned promotion to and from the 2. Bundesliga or were relegated from the league:


2. Bundesliga, Records, Player records, Top scorers
Boldface indicates a player still active in the 2. Bundesliga.


2. Bundesliga, Spectators
The spectator figures since 1992, when the league returned to the single division format:


2. Bundesliga, Second division in the European Cup
So far, clubs of the 2. Bundesliga have participated in the European Cup six times:


2. Bundesliga Nord (1974–1981)
The 2. Bundesliga Nord was the second-highest level of the West German football league system in the north of West Germany from its introduction in 1974 until the formation of the single-division 2. Bundesliga in 1981. It covered the northern states of North Rhine-Westphalia, Lower Saxony, Bremen, Hamburg, Schleswig-Holstein and the city of West Berlin.


2. Bundesliga Nord (1974–1981), Overview
The league was established in 1974 to reduce the number of second divisions in Germany from five to two and thereby allow direct promotion to the league winners. Along with the foundation of the 2. Bundesliga Nord, formed from clubs of the three former Regionalligas of Nord, Berlin and West, went the foundation of the 2. Bundesliga Süd, which was created from clubs of the other two Regionalligas, Süd and Südwest.


2. Bundesliga Nord (1974–1981), Overview
The league was created from twelve clubs from the West, seven from the North and one from Berlin, reflecting the playing strength of the old Regionalligas.


2. Bundesliga Nord (1974–1981), Overview
The winner of the 2. Bundesliga Nord was directly promoted to the Bundesliga, the runners-up played a home-and-away series versus the southern runners-up for the third promotion spot.


2. Bundesliga Nord (1974–1981), Overview
The bottom four teams were relegated to the Amateurligas, after 1978 to the new Oberligas. In 1979 only three teams were relegated. The winners of the Amateurligas/Oberligas had to determine the promoted teams by the way of a promotion play-off.


2. Bundesliga Nord (1974–1981), Overview
The league operated with 20 teams in its first six seasons, expanding to 22 in its last in 1981.


2. Bundesliga Nord (1974–1981), Overview
Until 1978, below the 2. Bundesliga Nord ranked the following Verbandsligas and Oberligas:


2. Bundesliga Nord (1974–1981), Overview
In 1981, the two 2. Bundesligas merged into one, country-wide division. Nine clubs from the south and eight from the north plus the three relegated teams from the Bundesliga were admitted into the new league, the 2. Bundesliga.


2. Bundesliga Nord (1974–1981), Overview
SG Wattenscheid 09, VfL Osnabrück, Alemannia Aachen, SC Fortuna Köln and SC Preußen Münster all played every of the seven seasons of the league.


2. Bundesliga Nord (1974–1981), Disbanding of the 2. Bundesliga Nord
The league was dissolved in 1981. Eight clubs of the league went to the new 2. Bundesliga while the champion and runners-up were promoted to the Bundesliga. The twelve remaining clubs were relegated to the Amateurligas.


2. Bundesliga Nord (1974–1981), Play-offs for Bundesliga promotion
The third promotion spot to the Bundesliga was decided through a play-off round of the runners-up of the two 2nd Bundesligas. Here are the results of this round:


2. Bundesliga Süd (1974–81)
The 2. Bundesliga Süd was the second-highest level of the West German football league system in the south of West Germany from its introduction in 1974 until the formation of the single-division 2. Bundesliga in 1981. It covered the southern states of Saarland, Rheinland-Pfalz, Baden-Württemberg, Hesse and Bavaria.


2. Bundesliga Süd (1974–81), Overview
The league was established in 1974 to reduce the number of second divisions in Germany from five to two and thereby allow direct promotion to the league winners. Along with the foundation of the 2. Bundesliga Süd, formed from clubs of the two former Regionalligas of Süd and Südwest, went the foundation of the 2. Bundesliga Nord, which was created from clubs of the other three Regionalligas, Nord, Berlin and West.


2. Bundesliga Süd (1974–81), Overview
The league was created from thirteen southern and seven southwestern clubs, reflecting the difference in size of the two regions, south being much the larger.


2. Bundesliga Süd (1974–81), Overview
The winner of the 2. Bundesliga Süd was directly promoted to the Bundesliga, the runners-up played a home-and-away series versus the northern runners-up for the third promotion spot.


2. Bundesliga Süd (1974–81), Overview
The league operated with 20 teams in six seasons of its existence, only in 1980 was there 21 teams in the league. The bottom three, some years four teams were relegated to the Amateurligas, after 1978 to the new Oberligas.


2. Bundesliga Süd (1974–81), Overview
Until 1978, below the 2. Bundesliga Süd ranked the following Amateurligas:


2. Bundesliga Süd (1974–81), Overview
The winners of the larger leagues of Bayern and Hessen were directly promoted while the other seven leagues had to play-off for two more promotion spots. After 1978, these seven leagues merged down to two new leagues and the champions of those four remaining leagues, now called Oberligas, were all directly promoted to the 2. Bundesliga Süd.


2. Bundesliga Süd (1974–81), Overview
In 1981, the two 2. Bundesligas merged into one, country-wide division. Nine clubs from the south and eight from the north plus the three relegated teams from the Bundesliga were admitted into the new league, the 2. Bundesliga.


2. Bundesliga Süd (1974–81), Overview
The Stuttgarter Kickers, SV Waldhof Mannheim, SpVgg Fürth, SpVgg Bayreuth and FC Homburg all played every one of the seven seasons of the league.


2. Bundesliga Süd (1974–81), Qualifying to the 2. Bundesliga Süd
From the Regionalliga Südwest, seven clubs qualified for the new 2. Bundesliga Süd, from the Regionalliga Süd it was 13 clubs.


2. Bundesliga Süd (1974–81), Qualifying to the 2. Bundesliga Süd
The qualifying modus saw the last five seasons counted, whereby the last placed team in each season received one point, the second-last two points and so on. For a Bundesliga season within this five-year period, a club received 25 points, for an Amateurliga season none.


2. Bundesliga Süd (1974–81), Qualifying to the 2. Bundesliga Süd
For the seasons 1969–70 and 70–71, the received points counted single, for the 71–72 and 72–73 season double and for the 73–74 season three times.


2. Bundesliga Süd (1974–81), Qualifying to the 2. Bundesliga Süd
To be considered in the points table for the new league, a club had to play either in the Regionalliga in 1973–74 or to have been relegated from the Bundesliga to it for the next season, something which did not apply for the south as both teams relegated from the Bundesliga in 1974 went to the north.


2. Bundesliga Süd (1974–81), Qualifying to the 2. Bundesliga Süd
The bottom three clubs in each league, nominally the relegated teams in every other season, were barred from entry to the 2. Bundesliga, regardless of where they stood in the points ranking.


2. Bundesliga Süd (1974–81), Disbanding of the 2. Bundesliga Süd
The league was dissolved in 1981. According to their performance of the last couple of seasons, nine clubs of the league went to the new 2. Bundesliga while the champion was promoted to Bundesliga. The ten remaining clubs were relegated to the Amateurligas.


2. Bundesliga Süd (1974–81), Disbanding of the 2. Bundesliga Süd
Of the nine clubs, only one came from the southwest region, VfR Wormatia Worms, all others were southern clubs.


2. Bundesliga Süd (1974–81), Disbanding of the 2. Bundesliga Süd
The reduction in numbers of second division teams hit especially the Saarland hard, having their three most well known clubs, all members of the Bundesliga at some stage, relegated. While the FC Homburg and the 1. FC Saarbrücken soon bounced back, Borussia Neunkirchen never returned to second division football but did remain a force in the Oberliga Südwest.


2. Bundesliga Süd (1974–81), Play-offs for Bundesliga promotion
The third promotion spot to the Bundesliga was decided through a play-off round of the runners-up of the two 2. Bundesligas. Here are the results of this round:


2. Divisioona
2. Divisioona is the fourth highest level of ice hockey in Finland. Teams playing in 2. Divisioona can be relegated to 3. Divisioona or promoted to Suomi-sarja. From 1975 to 1999, 2. Divisioona was the third highest level of ice hockey in Finland before being replaced by Suomi-sarja.


2. Frauen-Bundesliga
The 2. Frauen-Bundesliga is the second league competition for women's association football in Germany. For its first 14 seasons the league was divided into two groups: Nord and Süd. The winner and the runner-up are promoted to the Bundesliga (unless they are reserve teams of Bundesliga sides); the last three places are relegated to the Regionalliga. Until the 2017–18 season, in each group, the winner was promoted and the bottom two were relegated.


2. Frauen-Bundesliga
The league has been played as one group of 14 teams since the 2018–19 season, with second teams of clubs being ineligible for promotion and allowed to have only three players older than 20 years.


2. Frauen-Bundesliga
For the 2020–21 season only, the 2. Frauen-Bundesliga was divided into two groups of 10 and nine teams each due to the relegation being suspended for the 2019–20 season as a result of COVID-19 pandemic. The two group winners will be promoted to the Frauen-Bundesliga for the 2021–22 season.


2. Handball-Bundesliga
The 2. Handball-Bundesliga is the second tier of professional handball in Germany.


2. Handball-Bundesliga, Season
It is directly linked to the Handball-Bundesliga, the country's highest tier, by a promotion and relegation system. Before the 2011–12 season, the league played in two regional groups (north and south), since then it has been playing in a nationwide single division format.


2. Handball-Bundesliga, Relegation and promotion
The top two placed teams are promoted to the Handball-Bundesliga for the next season. The five last placed teams are relegated to the 3. Liga.


2. Hannoversches Infanterie-Regiment Nr. 77
2. Hannoversches Infanterie-Regiment Nr. 77 (2nd Hanoverian Infantry Regiment Nr. 77) was an active infantry regiment in the Prussian Army (1867 to 1871) and German Imperial Army (1871 to 1918).


2. Liga (Austria)
The Austrian Football Second League (German: 2. Liga) is the second-highest professional division in Austrian football. It was formerly called the First League (Erste Liga), from 2002 to 2018.


2. Liga (Austria)
The division currently contains 16 teams, and the champion of the league is promoted to the Austrian Bundesliga. The three last placed teams are directly relegated from the Second League into the regional leagues. The Austrian Football Second Bundesliga is currently known as HPYBET 2. liga for sponsorship reasons.


2. Liga (Austria), Teams and stadia for the 2021–22 season
Starting in the 2018–19 season, the former First League changed its name to the Second League and expanded from 10 teams to 16 teams.


2. Liga (Austria), Teams and stadia for the 2021–22 season
The 16 teams competing in the 2020–21 Second League season are:


2. Liga (Austria), Teams
Sixteen teams will participate in the 2021–22 season. The only added team is St. Pölten, relegated from the 2020–21 Austrian Football Bundesliga.


2. Liga (Austria), Teams
Due to the suspension of the 2020–21 Austrian Regionalliga, no club was relegated from last season


2. Liga (Austria), Relegation
The destination of a club relegated from the Second League depends upon which Land (state) of the Federal Republic it is a member. The relegated clubs join one of the Regionalligen (regional leagues) in the east, centre or west of the country. The three regional league champions are promoted to the Second League. Participation in the professional Second League is conditional on their licensing by the fifth senate of the federal league. If the licence is refused for economic reasons, one team fewer will be relegated.


2. Liga (Austria), Name history
The Austrian second division has had several different names and sponsors since 1974.


2. Liga (Austria), Name history
(Seasons below represent the first season when the name was used)


2. Liga (Austria), Name history
The league was known as the Sky Go Erste Liga for sponsorship reasons from 2014/15 to 2017/18, but Sky is not mentioned on the official website 2liga.at, or in the ÖFB's 2018/19 preview articles.


2. Liga (Slovakia)
The 2. Liga is the second-level football league in Slovakia and is an amateur league. Currently, there are sixteen teams in one group of the competition. For two seasons, it was also known as DOXXbet liga for sponsorship reasons.


2. Liga (Slovakia), History
The league was formed as a second-tier league in Czechoslovakia. Before the dissolution of Czechoslovakia it consisted of 16 teams. Upon dissolution, six teams were promoted to the then newly formed Corgoň Liga. The league was expanded to 18 teams in 1996-97 season, but back to 16 in 2001-02 and 12 in 2006-07.


2. Liga (Slovakia), History
For 2014/2015 & 2015/2016 seasons the league became known as DOXXbet liga as part of a sponsorship agreement.


2. Liga (Slovakia), Winners
1FC VSS Košice did not meet club license rules and they went into bankruptcy


2. Liga (Slovakia), Performance by club
1- Inter Bratislava won league in 2008-09, but license was sold to FK Senica. 2- FC VSS Košice won league in 2016–17, but did not meet club license rules and they went into bankruptcy


2. Liga (Switzerland)
2. Liga is the sixth tier of the Swiss football league system. The division is split into 17 groups of 12, 13 or 14 teams by geographical region. Teams usually play within their own canton.


2. Liga Interregional
2. Liga Interregional is the fifth tier of the Swiss football league system. From 2000 to 2012, it was the country's fourth level. The division is split into 6 groups of 14 teams, by geographical region.


2. Liga Interregional, Current Season
The 2019–20 season is the current season in the Swiss 2. Liga Interregional.


2. OG
2. OG is the second and last studio album by German boy band Overground. It was released on November 8, 2004 by Cheyenne, Polydor and Zeitgeist.


2. Oberliga Süd
The 2. Oberliga Süd was the second-highest level of the German football league system in the south of Germany from 1950 until the formation of the Bundesliga in 1963. It covered the three states of Bavaria, Baden-Württemberg and Hesse.


2. Oberliga Süd, Overview
The 2. Oberliga Süd was formed in 1950 as a feeder league to the Oberliga Süd which had been operating since 1945. It was the second of the three second Oberligas, the other two being 2. Oberliga West (1949) and 2. Oberliga Südwest (1951).


2. Oberliga Süd, Overview
The winners and runners-up of this league were promoted to the Oberliga Süd, the bottom two teams relegated to the Amateurligas. Below the 2nd Oberliga were the following Amateurligas:


2. Oberliga Süd, Overview
The 1. FC Pforzheim was the only club to have played all 13 seasons in the league.


2. Oberliga Süd, Disbanding of the 2. Oberliga
In 1963 the league was dissolved. The teams on the places one to nine went to the new Regionalliga Süd, the new second division. The teams from 10 to 18 were relegated to the Amateurligas.


2. Oberliga Südwest
The 2. Oberliga Südwest (English: 2nd Premier league Southwest) was the second-highest level of the German football league system in the southwest of Germany from 1951 until the formation of the Bundesliga in 1963. It covered the two states of Rhineland-Palatinate and Saarland.


2. Oberliga Südwest, Overview
The 2. Oberliga Südwest was formed in 1951 as a feeder league to the Oberliga Südwest which had been operating from 1945. It was the last of the three second Oberligas, the other two being 2. Oberliga West (1949) and 2. Oberliga Süd (1950).


2. Oberliga Südwest, Overview
The winners and runners-up of this league were promoted to the Oberliga Südwest, the bottom two teams relegated to the Amateurligas. Below the 2. Oberliga were the following Amateurligas:


2. Oberliga Südwest, Overview
The TSC Zweibrücken was the only club to play all twelve seasons in the league.


2. Oberliga Südwest, Disbanding of the 2. Oberliga
In 1963 the league was dissolved. Six of its clubs went to the new Regionalliga Südwest, the new second division. The other ten teams were relegated to the Amateurligas.


2. Oberliga West
The 2. Oberliga West (English: 2nd Premier league West) was the second-highest level of the German football league system in the west of Germany from 1949 until the formation of the Bundesliga in 1963. It covered the state of North Rhine-Westphalia, the most populous state of Germany.


2. Oberliga West, Overview
The 2. Oberliga West was formed in 1949 with two groups of sixteen teams each, as a feeder league to Oberliga West. It was the first of the three 2nd Oberligas, the other two being 2. Oberliga Süd (formed in 1950) and 2. Oberliga Südwest (in 1951). The league adopted a single group format starting from 1952: The top two teams of the 2. Oberliga gained promotion to the Oberliga while the bottom two teams were relegated to the Verbandsliga's. Some years, however, exceptions applied and the number of promoted and relegated teams altered.


2. Oberliga West, Overview
In July 1955 the Westdeutscher Fußballverband decided to disband the league from 1956, but the German federation DFB outlawed this decision and the league continued its existence.


2. Oberliga West, Overview
Below the 2. Oberliga West ranked the Amateurliga's, varying in numbers (and names) but split into three zones, Niederrhein, Mittelrhein, and Westfalen :


2. Oberliga West, Overview
The 2. Oberliga West existed until 1963, when it was replaced by the Regionalliga West as the second division for North Rhine-Westphalia.


2. Oberliga West, Overview
SpVgg Herten is the only team to have played in the league for all 14 seasons.


2. Oberliga West, Overview
Schwarz-Weiß Essen became the first second-division side to win the German Cup when they did so in 1959, an achievement later repeated by Kickers Offenbach and Hannover 96 in the history of German football.


2. Oberliga West, Disbanding of the 2. Oberliga
In 1963, the league was disbanded in favor of the new Regionalliga. The first eight teams from this season went to the Regionalliga West. The clubs placed from ninth to sixteenth were relegated to the Verbandsligas.


2. Rugby Bundesliga (Austria)
The 2. Rugby Bundesliga is the second level of domestic club rugby union competition in Austria. Wombats RC of Wiener Neustadt is the only separate club in the league, the rest of the teams being the second teams of the 1. Rugby Bundesliga clubs.


2. Rugby Bundesliga (Austria), History
The 2. Rugby Bundesliga was first played in the 1995/1996 season.


2. Rugby Bundesliga (Austria), Results
The scores in blue are links to accounts of finals on the site of the Austrian Rugby Federation (ÖRV) - in German


2. Rugby-Bundesliga
The 2. Rugby-Bundesliga is the second-highest level of Germany's Rugby union league system, organised by the German Rugby Federation. Its set below the Rugby-Bundesliga, the top-tier of German rugby, and above the Rugby-Regionalliga, the third tier.


2. Rugby-Bundesliga
The league was undergoing a major revamp for the 2012-13 season with the number of clubs expanded from 20 to 24 and the league divided into four regional divisions of six teams each. Above the 2nd Bundesliga the Bundesliga was organised in a similar fashion.


2. Rugby-Bundesliga, History
The Second Bundesligas were formed after the German reunion in the early 1990s, originally as four regional leagues, North, East, West and South. Later, the four leagues were merged to form two leagues, South/West and North/East. By 2017 the league structure had been again divided into four groups.


2. Rugby-Bundesliga, History
Since 2002, the two champions of the leagues play a 2nd Bundesliga championship final which also determines the clubs promoted to the Bundesliga.


2. Rugby-Bundesliga, History
The bottom clubs in the league are relegated to the Rugby-Regionalliga, the third tier of rugby in Germany. The top-teams of the Regionalliga are promoted in turn.


2. Rugby-Bundesliga, History
In the 2008-09 season, only nine of the sixteen German states have clubs at this level. Bremen, Mecklenburg-Vorpommern, Saxony, Saxony-Anhalt, Schleswig-Holstein, Thuringia and Saarland have no 2nd Bundesliga club.


2. Rugby-Bundesliga, History
With the RC Luxembourg, a team from Luxembourg, a non-German side competes in the league. The inclusion of this side in the 2009 promotion round to the 2nd Bundesliga caused some debate in regards to its legality, as the club had not qualified through the German league system and was not a member of any of the German regional rugby federations. However, RC Luxembourg's application was declared valid in regards to the German rugby federations rules and regulations and the team finished second in the promotion round, earning a place in the 2nd Bundesliga for 2009-10.


2. Rugby-Bundesliga, History
As a sign of the gap between the Bundesliga and the 2nd Bundesliga, TSV Victoria Linden, who only won the 2009-10 North/East division in the last round of the championship, declined promotion, citing the additional cost of travelling and the limited player pool as their reason. The South/West champion, Stuttgarter RC, has also indicated that it would not take up promotion, leaving the Bundesliga with only eight clubs for the next season. It also meant, for the first time ever, that no club from Hanover would compete at the top level of German rugby.


2. Rugby-Bundesliga, History
DSV 78 protested the decision to reduce the league to eight teams again and thereby relegating the club. For the 2nd Bundesliga, this also meant, only one team each would be promoted to the two regional divisions to keep the strength at ten teams per league. In the North/East, this was SC Siemensstadt, the club being directly promoted, while, in the South/West, TV Pforzheim won the promotion tournament of the five southern Regionalliga champions in Nuremberg on 12 June 2010.


2. Rugby-Bundesliga, History
On 22 August 2011, five days before the 2011-12 season start, the RG Heidelberg withdrew its reserve team from the 2nd Rugby-Bundesliga South/West, citing inability to guarantee a full player squad for the whole season and thereby forcing the South/West division to compete with nine clubs only. The Berliner RC II withdrew during the season, also citing a lack of players as the reason while the RU Hohen Neuendorf was disqualified after not fielding a team in two league games.


2. Rugby-Bundesliga, History
In mid-July 2012 the Deutsche Rugby Tag, the annual general meeting of the DRV decided to approve a league reform proposed by German international Manuel Wilhelm. The new system saw the number of clubs in the 2nd Bundesliga increased from 20 to 24 and the league divided into four regional divisions of six clubs each. One of the main aims of the reform was to reduce the number of kilometres travelled by individual teams and therefore reduce the travel expenses. The system will remain mostly unchanged for the 2013-14 season.


2. Rugby-Bundesliga, History
The only changes will be a play-off between the fourth and fifth placed teams in each group after the first stage to determine the clubs advancing to the second stage. The championship play-offs after the second stage will be reduced from sixteen to twelve clubs with the top two teams in each group advancing directly to the quarter finals while the remaining eight will play a wild card round to determine the other four quarter finalists.


2. Rugby-Bundesliga, History
From 2015 onwards the Bundesliga was reduced to sixteen clubs while the 2. Bundesliga remains at twenty four, divided into four regional groups. The winners of these four divisions would play each other for two direct promotion spots to the Bundesliga while the two losers of these games would face the seventh placed clubs from the Bundesliga.


2. Rugby-Bundesliga, Clubs Participating in 2018
•  • DRC Hannover •  • FC St. Pauli • TSV Victoria Linden


2. Rugby-Bundesliga, Clubs Participating in 2018
• Berliner RC II •  •  •  • USV Potsdam •


2. Rugby-Bundesliga, Clubs Participating in 2018
•  • RC Aachen • RC Luxembourg •  •  • SC Frankfurt 1880 II •  •


2. Rugby-Bundesliga, Clubs Participating in 2018
• Heidelberger TV • Karlsruher SV • München RFC • RC Rottweil • RC Unterföhring • StuSta München • Stuttgarter RC


2. Rugby-Bundesliga, 2nd Bundesliga champions until 2001
Before the establishment of the single-division Bundesliga, the two 2nd Bundesligas determined their champion in the autumn half of the season before splitting each division into two groups, the upper half playing for Bundesliga promotion together with the bottom clubs of the Bundesliga, and the lower half against 2nd Bundesliga relegation. Teams from the two regional divisions did not meet and now country-wide 2nd Bundesliga championship as such was played. The autumn champions were:


2. Rugby-Bundesliga, Placings in the 2nd Bundesliga since 2001
Since 1998, the following clubs have played in the league. From 1999 to 2001, the league consisted of two regional divisions of six teams each. After an autumn (A) round, the top three from each group would reach the championship finals round in spring (S). The bottom three, together with the top three from each of the two 2nd Bundesligas would play a promotion round in spring with the top three in each group playing in the Bundesliga the following autumn. In 1997-98 and from 2001 to 2012, the league has been played in a single-division format. From 2012–13 to 2014–15 it had been divided into a first (I) and second round (II), followed by play-offs:


2. SKL
The Slovenian Second Basketball League (Slovene: 2. slovenska košarkarska liga), abbreviated as the 2. SKL, is the second-highest basketball league in Slovenia.


2. SKL, Format
Each team plays with each other team of the division twice, once at home and once at their opponent's arena.


2. deild
2. deild is the third tier league of football in the Faroe Islands. It was founded in 1976 and is organized by the Faroe Islands Football Association. It was originally called 3. deild but became 2. deild after a reorganization of the Faroe Islands football league system in 2005.


2. deild, History
The league was founded in 1943 and was named Meðaldeildin (The Middle Division): this was 1 year after the founding of the  Meistaradeildin, the original top level Faroe Islands football league. There was no promotion and relegation system at the time, and the two leagues operated separately from each other. This was partly because Meðaldeildin was mostly made up of B teams from the clubs in the Meistaradeildin. One year after all leagues had to be suspended due to the British occupation of the Faroe Islands. The league then resumed in 1945.


2. deild, History
From 1943 to 1975 it was called Meðaldeildin, then the league structure changed in 1976. The Meistaradeildin was renamed 1. deild and the Meðaldeildin was renamed 2. deild. Also from that season onwards, the top clubs in 2. deild were promoted to 1. deild. The first club to be promoted was Fram Tórshavn after they won the division for the first time in their history, though they finished bottom of 1. deild the following season and were relegated back into 2. deild. The latest change to the league was made in 2005, when 1. deild was renamed as Formuladeildin for sponsorship reasons; 2. Deild adopted the name of 1. Deild; the third tier was renamed 2. Deild and the fourth tier became 3. Deild.


2. deild, History
It currently has 10 participating teams. At the end of each season, two teams are relegated and two promoted from what is now the fourth division, pending the fact that the winning team in any given division doesn't already have a senior team in the division it is being promoted to. In such cases the team that finished second will be promoted in its stead. If a team is relegated to a division where one of its teams are already playing, the second best team will move one division down, thereby saving another team from relegation.


2. deild, History
At the end of the 2008 season, Fram, who were set to be renamed FC Hoyvík for the 2009 season and AB II finished in the  promotion places. Fram (now FC Hoyvík) were promoted to 1. deild, while AB II would only be promoted if their 1st team finished in one of the promotion places in 1. deild. AB's first team eventually won promotion into the Vodafonedeildin, finishing runners-up in 1.Deild. However, AB II were denied promotion to 1.deild and MB Midvágur who finished 3rd in 2.Deild, were promoted instead. Because AB II who finished 2nd, had used illegal players.


2. deild, History
Though there is still an ongoing debate whether they finished 2nd or 3rd and which team should have been promoted. The following season in 2009 MB Midvágur finished bottom in 1.Deild, only picking up 8 points and were relegated back into 2.deild. Meanwhile AB Argir II won the 2.deild in 2009 and were promoted to 1.deild.


2. deild, 2. deild seasons, Titles by team
In bold clubs currently playing in 2. deild. In italics the clubs that no longer exist.


2. deild karla
2. deild karla (English: Men's Second Division) is a football league in Iceland. It is the third division in the Icelandic football league system. The current champions are Afturelding from Mosfellsbær, who won their 1st title in 2018.


2. deild karla
The division was started in 1966 as third division with two provisional groups, with number of teams varying from 7 to 10 teams in each group. In 1987 it was decided to merge the two groups to form a single nationwide league with 10 teams so at the end of the 1987 season only three teams from each group would stay in the league, the top teams in each group were promoted, all other teams relegated to the 4th division and two teams promoted from the 4th division. In 1997 after a name change, the division became Second division.


2. deild karla
In 2008, as a part of a general change in Icelandic football, the number of teams was increased from 10 to 12.


2. deild karla (basketball)
2. deild karla (English: Men's 2. Division) or D2 is the third tier basketball competition among clubs in Iceland. It is organized by the Icelandic Basketball Federation (Icelandic: Körfuknattleikssamband Íslands – KKÍ). It consists of 11 teams and the season consists of a home-and-away schedule of 20 games. The top four non-reserve teams meet in a playoff for the victory in the league and promotion to 1. deild karla.


2. deild karla (basketball), History
The third tier league was founded in 1973 as 3. deild karla with 11 teams. In 1978 it was rebranded as 2. deild karla. On 13 March 2020, the 2019–20 season was postponed due to the coronavirus outbreak in Iceland. The day after, the Icelandic Basketball Federation canceled the rest of the season.


2. deild kvenna (football)
The 2. deild kvenna is the third tier women's football league in Iceland. The league was founded in 2017 and current champions are Augnablik. It features 8 teams and the top two qualify for a spot in the 1. deild kvenna.


2. divisjon
2. divisjon (often referred to as PostNord-ligaen for sponsorship reasons) is the third-highest level of the Norwegian football league system.


2. divisjon
There are 28 teams divided into two groups, and at the end of the season the winner of each group earns promotion to the second-highest division, 1. divisjon. The teams finishing in second place in their respective group will qualify for the promotion play-offs, where they will face each other. The winner will play against the 14th placed team in 1. divisjon for promotion. The bottom three teams in each group are relegated to 3. divisjon.


2. divisjon
2. divisjon is the highest league a reserve team can participate in, and only reserve teams from the Eliteserien clubs (first tier) are allowed to enter. The participation of reserve teams stirs debate from time to time.


2. divisjon, History
Between 1963 and 1990, 2. divisjon was the second highest level of the Norwegian football league system, therefore the name of the third highest level was 3. divisjon. When the highest level was rebranded in 1991, this level changed its name to 2. divisjon. From 2009 to 2011, the official name of the league was Fair Play ligaen, and from 2012 to 2015 the name was Oddsen-ligaen (after the main sponsor Norsk Tipping's betting-game called Oddsen). The league is currently branded as PostNord-ligaen, sponsored by PostNord.


2. divisjon, Current members
The following 28 clubs are competing in the 2021 2. divisjon:


2. divisjon, Winners, 1991–1995
All group winners, excluding second teams of top division teams, were promoted to 1. divisjon.


2. divisjon, Winners, 1996–2000
Each group winner played qualification play-offs to decide which teams promote to 1. divisjon. Teams in bold promoted to 1. divisjon through qualification play-offs.


2. divisjon, Winners, 2001–2016
All group winners, excluding second teams of top division teams, were promoted to 1. divisjon.


2. divisjon, Winners, 2017–
Teams in bold were promoted to 1. divisjon. Teams in italics were relegated to 2. divisjon.


2. divisjon, Reserve teams
Reserve teams of clubs from the two top divisions can participate in the 2. divisjon. Reserve teams of clubs from the 1. divisjon can not play in the 2. divisjon, so if a team is relegated from the 1. divisjon, the club's reserve team will be relegated to the 3. divisjon regardless of their final position in the league.


2. divisjon, Sponsorship
From 2016, 2. divisjon has its title sponsorship rights sold to PostNord.


2.-class torpedo boat
The 2.-class torpedo boat was a designation in the Scandinavian countries for a type of fast steam torpedo boats between 40 tons and 80 tons, in service from the 1880s to after World War I.