3,753 research outputs found
The Semigroups B\u3csub\u3e2\u3c/sub\u3e and B\u3csub\u3e0\u3c/sub\u3e are Inherently Nonfinitely Based, as Restriction Semigroups
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 �. For example, explicit bases of identities are found for the varieties generated by B0 and B2
Graph inverse semigroups: their characterization and completion
Graph inverse semigroups generalize the polycyclic inverse monoids and play
an important role in the theory of C*-algebras. This paper has two main goals:
first, to provide an abstract characterization of graph inverse semigroups; and
second, to show how they may be completed, under suitable conditions, to form
what we call the Cuntz-Krieger semigroup of the graph. This semigroup is the
ample semigroup of a topological groupoid associated with the graph, and the
semigroup analogue of the Leavitt path algebra of the graph.Comment: Some minor corrections made and tangential material remove
Identities in the Algebra of Partial Maps
We consider the identities of a variety of semigroup-related algebras modelling the algebra of partial maps. We show that the identities are intimately related to a weak semigroup deductive system and we show that the equational theory is decidable. We do this by giving a term rewriting system for the variety. We then show that this variety has many subvarieties whose equational theory interprets the full uniform word problem for semigroups and consequently are undecidable. As a corollary it is shown that the equational theory of Clifford semigroups whose natural order is a semilattice is undecidable
Inverse semigroups with idempotent-fixing automorphisms
A celebrated result of J. Thompson says that if a finite group has a
fixed-point-free automorphism of prime order, then is nilpotent. The main
purpose of this note is to extend this result to finite inverse semigroups. An
earlier related result of B. H. Neumann says that a uniquely 2-divisible group
with a fixed-point-free automorphism of order 2 is abelian. We similarly extend
this result to uniquely 2-divisible inverse semigroups.Comment: 7 pages in ijmart styl
Varieties of Restriction Semigroups and Varieties of Categories
The variety of restriction semigroups may be most simply described as that generated from inverse semigroups (S, ·, −1) by forgetting the inverse operation and retaining the two operations x+ = xx−1 and x* = x−1x. The subvariety B of strictrestriction semigroups is that generated by the Brandt semigroups. At the top of its lattice of subvarieties are the two intervals [B2, B2M = B] and [B0, B0M]. Here, B2and B0 are, respectively, generated by the five-element Brandt semigroup and that obtained by removing one of its nonidempotents. The other two varieties are their joins with the variety of all monoids. It is shown here that the interval [B2, B] is isomorphic to the lattice of varieties of categories, as introduced by Tilson in a seminal paper on this topic. Important concepts, such as the local and global varieties associated with monoids, are readily identified under this isomorphism. Two of Tilson\u27s major theorems have natural interpretations and application to the interval [B2, B] and, with modification, to the interval [B0, B0M] that lies below it. Further exploration may lead to applications in the reverse direction
Distributive inverse semigroups and non-commutative Stone dualities
We develop the theory of distributive inverse semigroups as the analogue of
distributive lattices without top element and prove that they are in a duality
with those etale groupoids having a spectral space of identities, where our
spectral spaces are not necessarily compact. We prove that Boolean inverse
semigroups can be characterized as those distributive inverse semigroups in
which every prime filter is an ultrafilter; we also provide a topological
characterization in terms of Hausdorffness. We extend the notion of the patch
topology to distributive inverse semigroups and prove that every distributive
inverse semigroup has a Booleanization. As applications of this result, we give
a new interpretation of Paterson's universal groupoid of an inverse semigroup
and by developing the theory of what we call tight coverages, we also provide a
conceptual foundation for Exel's tight groupoid.Comment: arXiv admin note: substantial text overlap with arXiv:1107.551
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