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"Algebraic structure" will mean a set with some n-ary operations defined on it. This does not include vector spaces for example.

During my study of algebra I have encountered mostly algebraic structures with one or two binary operations. I have not encountered any theory of structures with three or more. However, I do know examples of more operations being used in practice -- they just aren't included in the axioms, but defined in terms of the "main" operations.

Those implicit operations together with the axiomatized ones may or may not be interconnected by distributive laws. It seems to me the most natural of possible laws connecting two binary operations since it is related to the concept of homomorphism. Say I have a structure $(S,\star,\circ).$ To say that $\circ$ left-distributes over $\star$ means that for any $x,y,z\in S$

$$x\circ (y\,\star\, z)=(x\circ y)\star (x\circ z).$$

This means no less that for any $x\in S$ the function $\phi_x : S\longrightarrow S$ defined by the formula

$$\phi_x(s)=x\circ s$$

is an endomorphism of $(S,\star).$

I know axiomatized structures with two (or four if we want to count left and right distributive properties separately) distributive laws between distinct operations, that is distributive lattices. This is the most a structure with two operations can have. However, I have noticed that there are interesting examples of structures with more (however "implicit") distributive laws.


Is there any, even obscure, theory being developed which studies algebraic structures with more than two binary operations interconnected by more than two distributive laws (counting the distribution of $\circ$ over $\star$ only once, even if both left and right distributive laws hold)?


I am asking this question because I find it curious that even though such structures may appear in nature, I have not encountered any study of them. I will give two examples that have occured to me.

In both examples all laws will be two-sided. For "$\circ$ distributes over $\star$", I will write $$\circ\longrightarrow\star$$

A familiar example

Let $\mathbb N$ denote the set $\{1,2,\ldots\}.$ Let $+$ and $\cdot$ have their usual meanings. We have

$$\cdot\longrightarrow +$$

In $\mathbb N,$ we can define $\gcd$ and $\operatorname{lcm}$ to be the binary operations ascribing to a pair $(x,y)\in\mathbb N\times \mathbb N$ their positive greatest common divisor and positive least common multiple respectively. It is an exercise in elementary number theory to check that

$$\begin{eqnarray} \gcd\longrightarrow\operatorname{lcm}\\ \operatorname{lcm}\longrightarrow\gcd\\ \gcd\longrightarrow\gcd\\ \operatorname{lcm}\longrightarrow\operatorname{lcm}\\ \cdot\longrightarrow\gcd\\ \cdot\longrightarrow\operatorname{lcm} \end{eqnarray} $$

Together with $\cdot\longrightarrow +,$ this gives four binary operations and seven distributive laws.

An exotic example

Let $X$ be a set and $\mathscr B=2^{X\times X}$ be the set of all binary relations on $X.$ Since they are just elements of a power sets, binary relations can be unioned and intersected. Let's denote those operations in the standard way: $\cup$ and $\cap.$ It is known that

$$\begin{eqnarray} \cap\longrightarrow\cup\\ \cup\longrightarrow\cap \end{eqnarray} $$

For two binary relations $\rho,\sigma\in\mathscr B,$ we define

$$ \rho\circ\sigma=\left\{(x,y)\in X\times X\,|\,(\exists z\in X)\; (x,z)\in \rho\wedge (z,y)\in\sigma\right\}. $$

This is called the composition of binary relations. It is a well-known notion. Perhaps less known is the fact that

$$\circ\longrightarrow \cup$$

I'll omit the easy proof. We can notice that there seems to be place for a "dual" operation. Let's define

$$ \rho\star\sigma=\left\{(x,y)\in X\times X\,|\,(\forall z\in X)\; (x,z)\in \rho\vee (z,y)\in\sigma\right\}. $$

I do not know the name of this operation, nor do I know if it has ever been studied. It is probably easy to believe that the complement function $$':\mathscr B\longrightarrow \mathscr B$$ provides an isomorphism

$$(\mathscr B,\circ)\cong(\mathscr B,\star).$$

It may therefore be as easy to believe that


This gives us four binary operations on $\mathscr B$ interconnected by four distributive laws.

(This seems to be an extremely strange structure. I do have a question about it, although I'm not sure if it's a good idea to ask it here, as it's probably as far from anybody's interest as can be.)

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What's wrong with vector spaces (over a fixed field $F$) as algebraic structures? They have one binary operation (plus) and a bunch of unary operations (scalar multiplication by each element of $F$). –  Chris Eagle Mar 15 '12 at 19:41
@ChrisEagle I prefer them out of this question, because they are defined as algebraic structures in many differenct ways. For example, I've seen an awful definition which says that a vector space is a set $V\times F$ with the unnecessary (but defined) operations giving some "junk element" as their output. –  user23211 Mar 15 '12 at 19:45
GCD and LCM are just lattice operations (with BA atoms being primes powers). The algebra with (dot,gcd) signature is tropical semiring... –  Tegiri Nenashi Mar 15 '12 at 19:48
en.wikipedia.org/wiki/Relation_algebra#Definition (I admit, like in case of BA, this signature is somewhat arbitrary) –  Tegiri Nenashi Mar 15 '12 at 20:02
Distributivity laws, if you think about it, are really conditions which assert that one operation is a homomorphism with respect to another. This observation eventually leads to the idea of a commutative algebraic theory. –  Zhen Lin Mar 15 '12 at 20:03
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2 Answers

up vote 1 down vote accepted

Bilattices are examples of structures with more than 2 binary operations

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Thank you very much! –  user23211 Mar 15 '12 at 20:26
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Perhaps, algebras over operads is what you want http://en.wikipedia.org/wiki/Operad_theory.

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