# On the compactification of partial semigroups

We begin by introducing some relevant definitions.

Definition: A $\textit{partial semigroup}$ is a pair $(S,.)$ where $.$ maps a subset of $S \times S$ to $S$ and for all $a,b,c \in S, (a.b).c=a.(b.c)$ in the sense that if either side is defined, then so is the other and they are equal.

Definition : Let $(S, .)$ be a partial semigroup.

(a) For $x \in S, \phi(x)= \phi_S(x) = \{y \in S : x.y \; \mbox {is defined} \}.$

(b) The semigroup $S$ is $\textit{adequate}$ if and only if for every $F \in \mathcal{P}_f(S),$ the collection of finite subsets of $S,$ $\bigcap\limits_{x \in F} \phi(x) \neq \emptyset.$

(c) $\delta S = \bigcap\limits_{x \in S} cl_{\beta S} (\phi(x)),$ where $cl_{\beta S}(A)$ is the closure of $A$ with respect to the topology of $\beta S$, the Stone Cech compactification of $S.$

Definition: Let $T$ be a partial semigroup. Then $S$ is an $\textit{adequate partial subsemigroup}$ of $T$ if and only if $S \subseteq T, S$ is an adequate partial semigroup under the inherited operation, and for all $F \in \mathcal{P}_f(T)$ there exists $H \in \mathcal{P}_f(S)$ such that $\bigcap\limits_{x \in H} \phi_S(x) \subseteq \bigcap\limits_{x \in F} \phi_T(x).$

Definition: Let $S$ be a partial semigroup.

(a) A subset $I$ of $S$ is a left ideal of $S$ if and only if $x.y \in I$ whenever $x \in S$ and $y \in I \cap \phi(x).$

(b) A subset $I$ of $S$ is a right ideal of $S$ if and only if $x.y \in I$ whenever $x \in I$ and $y \in \phi(x).$

(c) A subset $I$ of $S$ is an ideal of $S$ if and only if $I$ is both a left ideal and a right ideal of $S.$

We begin by introducing some relevant definitions.

Let $E^{\diamond}$ be an adequate partial subsemigroup of $S^l,$ the product of $l$ copies of $S,$ with $\{(\overbrace{a,a,\cdots , a}^{l}) : a \in S\}\subseteq E^{\diamond}$ and $I^{\diamond}$ be the two sided adequate partial ideal of $E^{\diamond}.$

Define $I = \bigcap\limits_{\bar{x} \in I^{\diamond}} cl_{\beta S^l} \phi(\bar{x}).$

My questions are the following: 1) Is $I^{\diamond} \subseteq I.$ In particular, can we identify the elements of $I^{\diamond}$ by some ultrafilters from $I?$

2) How does $S,$ a partial semigroup, sit inside $\delta S?$