D-semigroups and constellations

Abstract

In a result generalising the Ehresmann-Schein-Nambooripad Theorem relating inverse semigroups to inductive groupoids, Lawson has shown that Ehresmann semigroups correspond to certain types of ordered (small) categories he calls Ehresmann categories. An important special case of this is the correspondence between two-sided restriction semigroups and what Lawson calls inductive categories.Gould and Hollings obtained a one-sided version of this last result, by establishing a similar correspondence between left restriction semigroups and certain ordered partial algebras they call inductive constellations. (A general constellation is a one-sided generalisation of a category.) We put this one-sided correspondence into a rather broader setting, at its most general involving left congruence D-semigroups (which need not satisfy any semiadequacy condition) and what we call co-restriction constellations, a finitely axiomatized class of partial algebras. There are ordered and unordered versions of our results.Two special cases have particular interest. One is that the class of left Ehresmann semigroups (the natural one-sided versions of Lawson's Ehresmann semigroups) corresponds to the class of co-restriction constellations satisfying a suitable semiadequacy condition. The other is that the class of ordered left Ehresmann semigroups (which generalise left restriction semigroups and for which semigroups of binary relations equipped with domain operation and the inclusion order are important examples) corresponds to a class of ordered constellations defined by a straightforward weakening of the inductive constellation axioms.

How to generalise demonic composition

“…Before moving on to determining exactly when the derived constellation of a demigroup is inductive, we pause to characterise those constellations that are derived constellations of demigroups. This would then provide the most general possible result along the lines of the results given in [7] (for left restriction semigroups) and then [18] (for D-semigroups). Since this is not our main focus, proofs (which follow the earlier cases fairly closely) are either briefly presented or omitted entirely, with a reference given to the earlier cases.…”

“…The proof of the following is formally exactly the same as the proof of Lemma 4.4 in [18]. Lemma 4.3 If P is a generalised co-restriction constellation, then for all e ∈ D(P ) and s ∈ P , e|e = e and (s|e) • e exists.…”

“…This is exactly how one defines the constellation product in a left restriction semigroup, but also for the more general types of unary semigroups as in [18].…”

“…In the previously considered less general cases, such a characterisation has been achieved via the notion of co-restriction on a constellation as in [18], and the same applies here. Compared to (R1)-(R5) in the definition of a co-restriction constellation in [17], where the derived constellations of D-semigroups were characterised, (GR1), and (GR2) are exactly (R1) and (R3) respectively, (GR3) fulfils a role similar to that of (R4) which is no longer assumed and also implies (R5), and no version of (R2) holds or is needed.…”

“…Likewise, the pattern of the following proof follows that of Proposition 4.5 in [18], but we include it here because there are some small differences. Proposition 4.4 Every generalised co-restriction constellation P determines a demigroup T(P ) with multiplication given by x ⊗ y := (x|D(y)) • y and with domain operation D the same as in P , and the derived constellation of T(P ) is P .…”