Peter R. Jones

Profinite categories, implicit operations and pseudovarieties of categories

Abstract The last decade has seen two methodological advances of particular direct import for the theory of finite monoids and indirect import for that of rational languages. The first has been the use of categories (considered as “algebras over graphs”) as a framework in which to study monoids and their homomorphisms, the second has been the use of implicit operations to study pseudovarieties of monoids. Still more recent work has emphasized the role of profiniteness in finite monoid theory. This paper fuses these three topics by means of a general study of profinite categories, with applications to C-varieties (pseudovarieties of categories) in general, to those C-varieties arising from M-varieties (pseudovarieties of monoids) in particular, to implicit operations on categories and to recognizable languages over graphs.

Locality of DS and associated varieties

We prove that the pseudovariety DS, of all finite monoids, each of whose regular D-classes is a subsemigroup, is local. (A pseudovariety (or variety) V is local if any category whose local monoids belong to V divides a member of V.) The proof uses the “kernel theorem” of the first author and Pustejovsky together with the description by Weil of DS as an iterated “block product”. The one-sided analogues of these methods provide wide new classes of local pseudovarieties of completely regular monoids. We conclude, however, with the second authors example of a variety (and a pseudovariety) of completely regular monoids that is not local.

Lattice isomorphisms of inverse semigroups

A largely untouched problem in the theory of inverse semigroups has been that of finding to what extent an inverse semigroup is determined by its lattice of inverse subsemigroups. In this paper we discover various properties preserved by lattice isomorphisms, and use these results to show that a free inverse semigroup ℱℐ x is determined by its lattice of inverse subsemigroups, in the strong sense that every lattice isomorphism of ℱℐ x upon an inverse semigroup T is induced by a unique isomorphism of ℱℐ x upon T. (A similar result for free groups was proved by Sadovski (12) in 1941. An account of this may be found in Suzukis monograph on the subject of subgroup lattices (14)).

Modular inverse semigroups

An inverse semigroup S is said to be modular if its lattice 𝓛𝓕 ( S ) of inverse subsemigroups is modular. We show that it is sufficient to study simple inverse semigroups which are not groups. Our main theorem states that such a semigroup S is modular if and only if (I) S is combinatorial, (II) its semilattice E of idempotents is “Archimedean” in S , (III) its maximum group homomorphic image G is locally cyclic and (IV) the poset of idempotents of each 𝓓-class of S is either a chain or contains exactly one pair of incomparable elements, each of which is maximal. Thus in view of earlier results of the second author a simple modular inverse semigroup is “almost” distributive. The bisimple modular inverse semigroups are explicitly constructed. It is remarkable that exactly one of these is nondistributive.

The lattice of full regular subsemigroups of a regular semigroup

Although the regular subsemigroups of a regular semigroup S do not, in general, form a lattice in any naturalway, it is shown that the full regular subsemigroups form a complete sublattice LF of the lattice of all subsemigroups; moreover this lattice has many of the nice features exhibited in (the special case of) the lattice of full inverse subsemigroups of an inverse semigroup, previously studied by one of the authors. In particular, LF is again a subdirect product of the corresponding lattices for each of the principal factors of S. A description of LF for completely 0-simple semigroups is given. From this, lattice-theoretic properties of LF may be found for completely semisimple semigroups. For instance, for any such combinatorial semigroup, LF is semimodular.

Semidistributive inverse semigroups

An inverse semigroup S is said to be meet (join) semidistributive if its lattice ( S ) of full inverse subsemigroups is meet (join) semidistributive. We show that every meet (join) semidistributive inverse semigroup is in fact distributive.

ON LATTICE ISOMORPHISMS OF INVERSE SEMIGROUPS

An L-isomorphism between inverse semigroups S and T is an isomorphism between their lattices L(S) and L(T ) of inverse subsemigroups. The author and others have shown that if S is aperiodic – has no nontrivial subgroups – then any such isomorphism Φ induces a bijection φ between S and T . We first characterize the bijections that arise in this way and go on to prove that under relatively weak ‘archimedean’ hypotheses, if φ restricts to an isomorphism on the semilattice of idempotents of S, then it must be an isomorphism on S itself, thus generating a result of Goberstein. The hypothesis on the restriction to idempotents is satisfied in many applications. We go on to prove theorems similar to the above for the class of completely semisimple inverse semigroups. 2000 Mathematics Subject Classification: Primary 20M18; Secondary 08A30 Over the past quarter-century, several authors have investigated the extent to which an inverse semigroup S is determined by its lattice L(S) of inverse subsemigroups (see the survey [8] and the monograph [12]): given an L-isomorphism, that is, an isomorphism Φ : L(S) → L(T ) for some inverse semigroup T , how are S and T related? It is easily seen that since Φ restricts to an L-isomorphism between their respective semilattices of idempotents, ES and ET , it induces a bijection φE between them. Following the lead of Goberstein [4] we focus here on the situation where φE is an isomorphism (see below for a rationale for this simplification). It has long been known that φE extends to a bijection φ : ES ∪NS → ET ∪NT , where NS denotes the set of elements that belong to no subgroup of S. In the aperiodic (or ‘combinatorial’) case where, by definition, all subgroups are trivial, φ is then a bijection between S and T . In turn, φ induces Φ in the obvious way. In this note we first characterize the bijections so obtained, in Theorem 2.3, and then in Theorem 4.3 find a general sufficient condition in order that this bijection should be an isomorphism, improving on some results of Goberstein [4].

Inverse Semigroups and their Lattices of Inverse Subsemigroups

For any class C of algebras it is natural to wonder how well algebras from C are determined by their lattices of subalgebras. This topic has a long history, beginning with subgroup lattices of groups (see §3). The subspace lattice of a vector space was shown to be intimately connected with projective geometry by R. Baer [1]. An interesting historical perspective may be found in the introduction to [33]. In my talk I will consider this topic from the point of view of inverse semigroups.

Varieties of restriction semigroups and varieties of categories

ABSTRACT 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 strict restriction 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, B2 and 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 Tilsons 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.

ON THE LATTICE OF FULL EVENTUALLY REGULAR SUBSEMIGROUPS

We generalize to eventually regular (or ‘π-regular’) semigroups the study of the lattice of full regular subsemigroups of a regular semigroup, which has its most complete exposition in the case of inverse semigroups. By means of a judicious definition, it is shown that the full eventually regular subsemigroups of such a semigroup form a complete lattice LF, which projects onto the lattices of full regular subsemigroups of its regular principal factors. Our deepest results are obtained for those eventually regular semigroups in which the regular elements form a subsemigroup. In that case, LF also projects onto the lattice of full regular subsemigroups of that regular subsemigroup. In particular, we characterize such semigroups for which LF is distributive. A much more explicit description is obtained for the eventually regular semigroups in which the idempotents commute.