Restriction Semigroups and Inductive Constellations

Gould, Victoria; Hollings, Christopher

doi:10.1080/00927870902887096

Cited by 40 publications

(98 citation statements)

References 21 publications

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“…An important example of a constellation is that arising from a left restriction semigroup, and these are characterised in [7] in terms of inductive constellations. However, much more general types of unary semigroups give rise to constellations in a similar way.…”

Section: D-semigroupsmentioning

confidence: 99%

“…In [7], Gould and Hollings showed how to define a one-sided version of a category they called a constellation that could be used to capture left restriction semigroups. As for two-sided restriction semigroups, one retains only some of the products in the left restriction semigroup S. But the lack of any notion of range means that more products must be retained.…”

Section: Introductionmentioning

confidence: 99%

“…A more general form of right multiplication by idempotent domain elements must also be retained, and as for two-sided restriction semigroups, only one partial order is needed (that given by a ≤ b if a = eb for some e ∈ D(S)). The corresponding class of ordered partial algebras is axiomatized in [7] as the class of inductive constellations. An isomorphism between the category of left restriction semigroups and the category of inductive constellations (with suitably defined arrows) is established there.…”

Section: Introductionmentioning

confidence: 99%

“…Aside from Ehresmann semigroups, examples of these include all binary relations on a set with domain restricted in some way (for example to finite subsets of an infinite set) but with ranges not so restricted. The approach in [7] does not seem capable of generalising to this setting, since a single partial order does not seem sufficient to capture both left and right multiplication by idempotents, whereas retaining two partial orders does not seem to work either because the right-sided partial order given by a ≤ b if and only if a = be for some domain element e ∈ D(S) is not compatible with the constellation product.…”

Section: Introductionmentioning

confidence: 99%

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D-semigroups and constellations

Stokes

2017

Semigroup Forum

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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.

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Section: D-semigroupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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D-semigroups and constellations

Stokes

2017

Semigroup Forum

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“…(We take this particular definition from [12], where it is attributed to Jackson and Stokes [15].) The last of each set of four identities are often termed the 'ample' identities.…”

Section: Introductionmentioning

confidence: 99%

THE SEMIGROUPS B₂AND B₀ARE INHERENTLY NONFINITELY BASED, AS RESTRICTION SEMIGROUPS

Jones

2013

Int. J. Algebra Comput.

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The five-element Brandt semigroup B2 and its four-element subsemigroup B0, obtained by omitting one nonidempotent, have played key roles in the study of varieties of semigroups. Regarded in that fashion, they have long been known to be finitely based. The semigroup B2 carries the natural structure of an inverse semigroup. Regarded as such, in the signature {⋅, -1}, it is also finitely based. It is perhaps surprising, then, that in the intermediate signature of restriction semigroups — essentially, "forgetting" the inverse operation x ↦ x-1 and retaining the induced operations x ↦ x+ = xx-1 and x ↦ x* = x-1x — it is not only nonfinitely based but inherently so (every locally finite variety that contains it is also nonfinitely based). The essence of the nonfinite behavior is actually exhibited in B0, which carries the natural structure of a restriction semigroup, inherited from B2. It is again inherently nonfinitely based, regarded in that fashion. It follows that any finite restriction semigroup on which the two unary operations do not coincide is nonfinitely based. Therefore for finite restriction semigroups, the existence of a finite basis is decidable "modulo monoids". These results are consequences of — and discovered as a result of — an analysis of varieties of "strict" restriction semigroups, namely those generated by Brandt semigroups and, more generally, of varieties of "completely r-semisimple" restriction semigroups: those semigroups in which no comparable projections are related under the generalized Green relation 𝔻. For example, explicit bases of identities are found for the varieties generated by B0 and B2.

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