section_path: Major branches/Universal algebra
section_name: Universal algebra
paragraph_text: Universal algebra
  
  Universal algebra is the study of algebraic structures in general. As part of its general perspective, it is not concerned with the specific elements that make up the underlying sets and considers operations with more than two inputs, such as ternary operations. It provides a framework for investigating what structural features different algebraic structures have in common. One of those structural features concerns the identities that are true in different algebraic structures. In this context, an identity is a universal equation or an equation that is true for all elements of the underlying set. For example, commutativity is a universal equation that states that a ∘ b {\displaystyle a\circ b} is identical to b ∘ a {\displaystyle b\circ a} for all elements. Two algebraic structures that share all their identities are said to belong to the same variety. For instance, the ring of integers and the ring of polynomials form part of the same variety because they have the same identities, such as commutativity and associativity. The field of rational numbers, by contrast, does not belong to this variety since it has additional identities, such as the existence of multiplicative inverses.
  
  Homomorphisms are tools in universal algebra to examine structural features by comparing two algebraic structures. A homomorphism is a function from the underlying set of one algebraic structure to the underlying set of another algebraic structure that preserves certain structural characteristics. If the two algebraic structures use binary operations and have the form ⟨ A, ∘ ⟩ {\displaystyle \langle A,\circ \rangle } and ⟨ B, ⋆ ⟩ {\displaystyle \langle B,\star \rangle } then the function h: A → B {\displaystyle h:A\to B} is a homomorphism if it fulfills the following requirement: h (x ∘ y) = h (x) ⋆ h (y) {\displaystyle h(x\circ y)=h(x)\star h(y)}. The existence of a homomorphism reveals that the operation ⋆ {\displaystyle \star } in the second algebraic structure plays the same role as the operation ∘ {\displaystyle \circ } does in the first algebraic structure. Isomorphisms are a special type of homomorphism that indicates a high degree of similarity between two algebraic structures. An isomorphism is a bijective homomorphism, meaning that it establishes a one-to-one relationship between the elements of the two algebraic structures. This implies that every element of the first algebraic structure is mapped to one unique element in the second structure without any unmapped elements in the second structure.
  
  Another tool of comparison is the relation between an algebraic structure and its subalgebra. The algebraic structure and its subalgebra use the same operations, which follow the same axioms. The only difference is that the underlying set of the subalgebra is a subset of the underlying set of the algebraic structure. All operations in the subalgebra are required to be closed in its underlying set, meaning that they only produce elements that belong to this set. For example, the set of even integers together with addition is a subalgebra of the full set of integers together with addition. This is the case because the sum of two even numbers is again an even number. But the set of odd integers together with addition is not a subalgebra because it is not closed: adding two odd numbers produces an even number, which is not part of the chosen subset.