Self-induced actions

It is common to represent a semigroup, monoid, or group by action on some set, but in the case where an action is not given to us we can recover one by the action of the semigroup on itself. This process can even be applied to magmas, to get the action of a magma on itself. In that case, the lack of associativity means that the self-induced actions need not be complete, which produces a subset of a semigroup. This leads to an interesting connection between semigroup theory and magma theory.

Left and right actions:

Introduction:
Let $S$ be a set, $* : S^2 \to S$ be a magma, and $T_{S}$ be the complete transformation monoid on $S$, whose members are actions $f: S \to S$. Then we can define the left and right actions of elements by mappings: \[ L : S \to T_{S} \] \[ L(a) = \lambda(x) (a*x) \] \[ R : S \to T_{S} \] \[ R(a) = \lambda(x) (x*a) \] The following elementary observations are immediate from these definitions:

• An element is L/R cancellative, unital, or zero if its corresponding L/R actions are mono, unital, or constant.
• The centralizer of an element is the equalizer of its left and right actions.
• An element is central if its left and right actions coincide.

Action systems:
The image of the L,R mappings are both subsets of $T_{S}$. For general magmas, these action systems $Im(L),Im(R)$ do not need to be semigroups, however, they are always subsets of a semigroup and partial semigroups. Their closures are the $L,R$ action semigroups.

Equal actions:
The L,R mappings don't need to be mono, instead they can produce equal actions for different values. This produces ${=}L, {=}R$ equivalence relations on $S$.

Example 1. in a constant semigroup all L,R actions are equal and so both the L,R mappings are constant.

[[1 1 1]
 [1 1 1]
 [1 1 1]]

Example 2. in the following pure rectangular band $L$ is mono and always maps to a different constant function and $R$ is constant and always maps to the same identity action.

[[1 1 1]
 [2 2 2]
 [3 3 3]]

In general semigroups there are often equal actions, as demonstrated in the above actions but that doesn't need to be the case. A magma is called left faithful or right faithful, provided that the left and right self actions $L,R$ are monomorphisms. Groups are clearly left and right faithful because if they had equal actions that would violate cancellativity.

The special case of semigroups:
In the special case of semigroups, $L,R$ are actually semigroup homomorphisms. Their image action systems are semigroups and the equivalence relations ${=}L,{=}R$ are congruences. This is clarified in the following theorem. By the first isomorphism theorem, we have isomorphisms $\frac{S}{{=}L} \cong Im(L)$ and $\frac{S}{{=}R} \cong Im(R)$.

Theorem. $L : S \to T_S$ and $R : S \to T_S^{Op}$ are semigroup homomorphisms.

Proof. By definition $L(ab) = \lambda(x) (ab)x$ by associativity $(ab)x = a(bx)$ but then $a(bx)$ is equal to $(L(a) \circ L(b))(x)$. Therefore, $L(ab) = L(a) \circ L(b)$. In other direction, $R(ab) = \lambda (x) x(ab)$. Then by associativity $x(ab) = (xa)b$. This is then equal to $(R(b) \circ R(a))(x)$. $R(ab) = R(b) \circ R(a)$ which is a homomorphism in the opposite direction.

Remarks. the distinction between $T_S$ and $T_S^{Op}$ doesn't really matter and is dependent upon our notation for composition. All that matters is that $L$ and $R$ are opposite directions, so their corresponding semigroups homomorphisms are opposite to one another.

In the case of semigroups, $L$ and $R$ can be seen as reducing semigroups to their faithful counterparts. The fact that group's are faithful directly leads to Cayley's theorem which is the representation of any group by the group action defined by $L$.

The special case of groups:
The action of a group on itself is simply transitive, which means that all elements are contained in the same orbit and all stabilizers are trivial. The former is a consequence of the existence of inverses and the later is a consequence of cancellativity.

Two sided actions:

The J semigroup:
The images $Im(L),Im(R)$ of the L and R mappings are both subsets of the semigroup $T_S$. Their closures are therefore both members of $Sub(T_S)$. They both must therefore have a semigroup join, which contains all actions both left and right.

Definition. the $J$ semigroup is the semigroup join of the $L$ and $R$ action semigroups, which are defined as the closures of the images of the $L$ and $R$ mappings.

$J$ is the ultimate semigroup defining actions of a magma, semigroup, or other related structure on itself.

Inclusions:
The three semigroups $L,R,J$ form a three element suborder of $Sub(T_S)$ with $L,R \subseteq J$. The intersection $L \cap R$ contains at least all actions by central elements.

Magma theory:

The $L,R$ mappings produce a connection between semigroup theory and magma theory that is worth examining. In particular, magmas correspond to subsets of semigroups, because their $L,R$ images don't need to be closed.

Example 1. consider the following non-commutative monogenic magma

[[2,1]
 [2,2]]

Then we have the following left actions: {[2,1],[2,2]}. This is a subset of $T_2$ but it is not closed, which demonstrates that the images of $L$ don't need to be closed for magmas. Its closure is ${[1,2],[2,1],[1,1],[2,2]}$ which is an upper invariant swapper semigroup because the half invariant upper constant elements are acted upon by their predecessor group. This is a first example of a semigroup constructed from a magma.

In the other direction we get the semigroup {[1,2],[2,2]} which is a subsemigroup, therefore this semigroup is R action composition closed. This semigroup action is the total order semilattice on two elements.

Example 2. let $\mathbb{Z}_+$ be the positive integers then $^ : \mathbb{Z}_+^2 \to \mathbb{Z}_+$ is a magma.

The right actions of exponentation are composition closed because $(a^x)^y = a^{xy}$. In this case, the operation is not associative so composition is determined by an operation other then exponentation itself. In this case, we have a semigroup homomorphism $Im(R) \to *$.

The left actions of exponentation on the other hand are not composition closed. When we have $a^(b^x)$ there is no way to reduce it to a single exponentation. Left actions are not composable at all unless one of them is equal to one, which is the trivial case. The semigroup closure of left actions is the set of all exponentation towers closed under composition.

As $Im(L)$ is a subset of a semigroup, we can form a Cayley digraph of it consisting of all pairs of elements $(a,b)$ such that there exists some $c$ such that $c^a = b$. We can call this the 'base of' binary relation, as elements are related provided that the first is a base of the other with respect to some exponent. This binary relation is not transitive because 3 is a base of 9, 9 is a base of 512, and 3 is not a base of 512.

Green's relations:

It is clear that Green's preorders is a special type of action preorder and now we can directly construct monoid actions whose action preorders correspond to the Green's preorders of a semigroup.

• $\subseteq_L$ is the action preorder of $Im(L)$
• $\subseteq_R$ is the action preorder of $Im(R)$
• $\subseteq_J$ is the action preorder of the $J$ semigroup

Green's relations are the strongly connected components of each of these action preorders. We can also form similar preorders from magmas or any structure that acts on itself in some manner.