On certain codes admitting inverse semigroups as syntactic monoids

The purpose of this paper is to investigate under what conditions an inverse semigroup M is isomorphic to the syntactic monoid M(A)* of afinite prefix code A over an alphabet X. We find a necessary condition for this to happen. It expresses a precise link between the group of units of M and the maximal subgroups of the 0-minimal ideal of M (Theorem 2.1). The condition is shown to be sufficient in case M is an ideal extension of a Brandt semigroup by a group (Corollary 2.3). We also introduce and study stable codes (products of subsets of the alphabet) and give structural properties of their syntactic monoids (Proposition 3.3 and Theorem 3.5). Most of our results inter-relate structural properties of certain semigroups and divisibility of integers attached to them. The terminology follows [1] and [3].     

Constructions and presentations for monoids

Presentations are found for the wreath product of two monoids, the Schützenberger product of two monoids, the Bruck-Reilly extension of a monoid, strong semilattices of monoids and Rees matrix semigroups of monoids.     

On presentations of Brauer-type monoids

We obtain presentations for the Brauer monoid, the partial analogue of the Brauer monoid, and for the greatest factorizable inverse submonoid of the dual symmetric inverse monoid. In all three cases we apply the same approach, based on the realization of all these monoids as Brauer-type monoids.     

Catalan monoids,monoids of local endomorphisms,and their presentations

The Catalan monoid and partial Catalan monoid of a directed graph are introduced. Also introduced is the notion of a local endomorphism of a tree, and it is shown that the Catalan (resp. partial Catalan) monoid of a tree is simply its monoid of extensive local endomorphisms (resp. partial endomorphisms) of finite shift. The main results of this paper are presentations for the Catalan and partial Catalan monoids of a tree. Our presentation for the Catalan monoid of a tree is used to give an alternative proof for a result of Higgins. We also identify results of Aîzen?tat and Popova which give presentations for the Catalan monoid and partial Catalan monoid of a finite symmetric chain.     

On algebraic semigroups and monoids,II

Consider an algebraic semigroup S and its closed subscheme of idempotents, E(S). When S is commutative, we show that E(S) is finite and reduced; if in addition S is irreducible, then E(S) is contained in a smallest closed irreducible subsemigroup of S, and this subsemigroup is an affine toric variety. It follows that E(S) (viewed as a partially ordered set) is the set of faces of a rational polyhedral convex cone. On the other hand, when S is an irreducible algebraic monoid, we show that E(S) is smooth, and its connected components are conjugacy classes of the unit group.     

Endomorphism monoids in varieties of commutative semigroups

A concrete category is almost universal if its class of non-constant morphisms contains an isomorphic copy of every category of algebras as a full subcategory. This paper characterizes almost universal varieties of commutative semigroups. As a consequence we obtain that for every infinite cardinal κ there exists a commutative semigroup of cardinality κ such that it has exactly two endomorphisms, the identity endomorphism and a single constant endomorphism.     

A construction of weakly inverse semigroups

Let S° be an inverse semigroup with semilattice biordered set E° of idempotents and E a weakly inverse biordered set with a subsemilattice Ep = { e ∈ E ｜ arbieary f ∈ E, S（f , e） loheain in w（e）} isomorphic to E° by θ：Ep→E°. In this paper, it is proved that if arbieary f, g ∈E, f ←→ g→→ f°θD^s° g°θand there exists a mapping φ from Ep into the symmetric weakly inverse semigroup P J（E∪ S°） satisfying six appropriate conditions, then a weakly inverse semigroup ∑ can be constructed in P J（S°）, called the weakly inverse hull of a weakly inverse system （S°, E, θ, φ） with I（∑） ≌ S°, E（∑） ∽- E. Conversely, every weakly inverse semigroup can be constructed in this way. Furthermore, a sufficient and necessary condition for two weakly inverse hulls to be isomorphic is also given.     

A relation for general and inverse semigroups

The relation in the title is S defined by

Generation of infinite factorizable inverse monoids

We investigate the generation of factorizable inverse monoids, paying special attention to the factorizable parts of the symmetric and dual symmetric inverse monoids. Key ideas covered include rank, relative rank, Sierpiński rank, and the semigroup Bergman property. The results for finite monoids are well-known or follow quickly from well-known facts, so most of the paper concerns the infinite case.     

Fundamental regular semigroups with inverse transversals

Let C be a regular semigroup with an inverse transversal C° and let C be generated by its idempotents. Following W. D. Munn and T. E. Hall’s idea, in this paper, a fundamental regular semigroup T _C,C° with an inverse transversal T _C,C° ^° is constructed such that the following holds. For any regular semigroup S with an inverse transversal S° and 〈E(S)〉 = C, C° = C ∩ S°, there is a homomorphism φ from S to T _C,C° such that the kernel of φ is the maximum idempotent-separating congruence on S, and φ satisfies: (1) φ| _C is a homomorphism from C onto 〈E(T _C,C°)〉 ; (2) φ| _S° is a homomorphism from S° to T _C,C° °. In particular, S is fundamental if and only if S is isomorphic to a full subsemigroup of T _C,C°. Our fundamental regular semigroup T _C,C° is isomorphic to a subsemigroup of the Hall semigroup of C but it is easier to handle. Its elements are partial transformations, and the operation—although not the usual composition—is defined by means of composition.