Subsets of semigroups

In this post, we will examine subsets of semigroups that are not necessarily subsemigroups. Let $S$ be a set, $* : S^2 \to S$ be a semigroup on $S$, and $T$ be a subset of $S$. Then we can form a restriction binary operation $*_{T^2} : T^2 \to S$, but this restricted binary operation may produce values that are not in $T$. We can restrict the semigroup further by a composability binary operation, which creates a partial operation.

Composability:
The composability binary relation of a subset $T$ of a semigroup has a pair of elements related to one another provided that their composition is in $T$.

Definition. let $(S,*)$ then composability $C(T) : \wp\{S\} \to \wp\{S^2\}$ is a map from subsets of $S$ to relations of $S$ defined by $\{ (x,y) \in T^2: x*y \in T \}$

By restricting to $C(T)$ instead of $T^2$ we get a partial binary operation because now it is defined on a subset of the cartesian product $T^2$, which is a partial binary operation.

Partial semigroups:
Let $* : R \to S$ be a partial binary operation with domain $R \subseteq S^2$. Suppose that we have an ordered triple $(x,y,z)$ of elements of $S$ then the compositions $(xy)z$ and $x(yz)$ need not exist. There is a nine-element existence lattice that determines the extent to which they do exist.

This defines the ontology of associatity conditions of partial binary operations. Given any existence condition, we can require that $(xy)z$ and $x(yz)$ exist and that they coincide.

• * is strongly associative if $xy, yz$ exists implies associativity
• * is left pre-associative if $(xy)z, yz$ exists implies associativity
• * is right pre-associative if $x(yz), xy$ exists implies associativity.
• * is a partial semigroup if $(xy)z, x(yz)$ exists implies associativity.

You could then form an ontology of conditions of associativity of partial binary operations by taking the ideals of the nine-element existence lattice. For further discussion about the ways of defining associativity of partial operations see [1]. Given a set $T$ with composability relation $C(T)$ we can now form a restricted binary operation $*_{C(T)} : C(T) \to T$ whose output set is $T$ rather then $S$. It is not hard too see that this is a partial semigroup because the parent operation is associative.

Proposition. $*_{C(T)} : C(T) \to T$ is a partial semigroup

It follows that we can form partial semigroups from any subset of a semigroup. A common technique is to take a partial semigroup and adjoin a zero element to it, so that all compositions that don't exist instead map to the zero element. This can be used to construct semigroups from certain partial semigroups. For example, given an ideal $I$ then its rees quotient is produced by adjoining a zero to the partial semigroup $I^C$.

Induced relations:
Recall that there is a monotone map from subsemigroups to action preorders. When we are dealing with general subsets, the lack of composition closure corresponds to a lack of transitivity. This leads to left/right Cayley relations $\Gamma$, which can be defined for any semigroup. Given a subset $T$ of a semigroup $(S,*)$ then $(x,y) \in \Gamma$ if $\exists t \in T: y = xt$. If the subset is composition closed, then this action relation is a preorder.

References:
[1] The Theory of Partial Algebraic Operations (Mathematics and Its Applications (414))