Finite state wreath powers of transformation semigroups

The finite state wreath power of a transformation semigroup is introduced. It is proved that the finite state wreath power of nontrivial semigroup is not finitely generated and in some cases even does not contain irreducible generating systems. The free product of two monogenic semigroups of index 1 and period m is constructed in the finite state wreath power of corresponding monogenic monoid.     

A generalization of the first Malcev theorem on nilpotent semigroups and nilpotency of the wreath product of semigroups

We describe all [0-]simple semigroups that are nilpotent in the sense of Malcev. This generalizes the first Malcev theorem on nilpotent (in the sense of Malcev) semigroups. It is proved that if the extended standard wreath product of semigroups is nilpotent in the sense of Malcev and the passive semigroup is not nilpotent, then the active semigroup of the wreath product is a finite nilpotent group. In addition to that, the passive semigroup is uniform periodic. Necessary and sufficient conditions are found under which the extended standard wreath product of semigroups is nilpotent in the sense of Malcev in the case where each of the semigroups of the wreath product generates a variety of finite step.     

The structure of L*-inverse semigroups

The concepts of L*-inverse semigroups and left wreath products of semigroups are introduced. It is shown that the L*-inverse semigroup can be described as the left wreath product of a type A semigroupΓand a left regular band B together with a mapping which maps the semigroupΓinto the endomorphism semigroup End(B). This result generalizes the structure theorem of Yamada for the left inverse semigroups in the class of regular semigroups. We shall also provide a constructed example for the L*-inverse semigroups by using the left wreath products.     

The structure of ?*-inverse semigroups

The concepts of ℒ*-inverse semigroups and left wreath products of semigroups are introduced. It is shown that the ℒ*-inverse semigroup can be described as the left wreath product of a type A semigroup Γ and a left regular band B together with a mapping which maps the semigroup Γ into the endomorphism semigroup End(B). This result generalizes the structure theorem of Yamada for the left inverse semigroups in the class of regular semigroups. We shall also provide a constructed example for the ℒ*-inverse semigroups by using the left wreath products.     

Nilpotent Ranks of Semigroups of Partial Transformations

A subset U of a semigroup S is a generating set for S if every element of S may be written as a finite product of elements of U. The rank of a finite semigroup S is the size of a minimal generating set of S, and the nilpotent rank of S is the size of a minimal generating set of S consisting of nilpotents in S. A partition of a q-element subset of the set X_n = {1,2,..., n} is said to be of type τ if the sizes of its classes form the partition τ of the positive integer q ≤ n. A non-trivial partition τ of q consists of k < q elements. For a non-trivial partition τ of q < n, the semigroup S(τ), generated by all the transformations with kernels of type τ, is nilpotent-generated. We prove that if τ is a non-trivial partition of q < n, then the rank and the nilpotent rank of S(τ) are both equal to the number of partitions X_n of type τ.     

Automorphisms of the endomorphism semigroup of a free monoid or a free semigroup

We determine all isomorphisms between the endomorphism semigroups of free monoids or free semigroups and prove that automorphisms of the endomorphism semigroup of a free monoid or a free semigroup are inner or ``mirror inner". In particular, we answer a question of B. I. Plotkin.

     

Endomorphisms of relatively free algebras with weak exchange properties

The structure of the endomorphism monoid of a stable basis algebra A is described. It is shown to be an abundant monoid; the subsemigroup of endomorphisms of finite rank has a regular semigroup of left quotients.     

The endomorphism monoid of a free trioid of rank 1

We describe all endomorphisms of a free trioid of rank 1 and construct a semigroup which is isomorphic to the endomorphism monoid of such free trioid. Also, we give an abstract characteristic for the endomorphism monoid of a free trioid of rank 1 and prove that free trioids are determined by their endomorphism monoids.     

图的字典序积和自同态幺半群

Ｆ．Ｈａｒａｒｙ￣［１］和Ｇ．Ｓａｂｉｄｕｓｓｉ￣［２］考虑过图Ｘ和ｙ的字典序积Ｘ［Ｙ］的自同构群ＡｕｔＸ［Ｙ］与它们各自的自同构群的圈积ＡｕｔＸ［ＡｕｔＹ］的关系，并给出了两者相等的一种刻划．在本文，我们考虑更广意义上的问题，即Ｘ［Ｙ］的自同态幺半群ＥｎｄＸ［Ｙ］与各自的自同态幺半群的圈积ＥｎｄＸ［ＥｎｄＹ］的关系，也给出了两者相等的一种刻划，同时得到了下面结果：如果Ｘ和Ｙ都是不含Ｋ＿３导出子图的连通图，且其中之一图有奇数围长，那么ＥｎｄＸ［Ｙ］＝ＥｎｄＸ［ＥｎｄＹ］．     

Affine stratifications from finite misère quotients

Given a morphism from an affine semigroup to an arbitrary commutative monoid, it is shown that every fiber possesses an affine stratification: a partition into a finite disjoint union of translates of normal affine semigroups. The proof rests on mesoprimary decomposition of monoid congruences and a novel list of equivalent conditions characterizing the existence of an affine stratification. The motivating consequence of the main result is a special case of a conjecture due to Guo and the author on the existence of affine stratifications for (the set of winning positions of) any lattice game. The special case proved here assumes that the lattice game has finite misère quotient, in the sense of Plambeck and Siegel.     

The minimal number of generators of a finite semigroup

The rank of a finite semigroup is the smallest number of elements required to generate the semigroup. A formula is given for the rank of an arbitrary (not necessarily regular) Rees matrix semigroup over a group. The formula is expressed in terms of the dimensions of the structure matrix, and the relative rank of a certain subset of the structure group obtained from subgroups generated by entries in the structure matrix, which is assumed to be in Graham normal form. This formula is then applied to answer questions about minimal generating sets of certain natural families of transformation semigroups. In particular, the problem of determining the maximum rank of a subsemigroup of the full transformation monoid (and of the symmetric inverse semigroup) is considered.     

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.     

Products of idempotent endomorphisms of free acts of infinite rank

In 1966, J. M. Howie characterized the self-maps of a set which can be written as a product (i.e., composite) of idempotent self-maps of that set. Using a wreath product construction introduced by V. Fleischer, the first-named author was recently able to describe products of idempotent endomorphisms of a freeS-act of finite rank whereS is any monoid. The purpose of the present paper is to extend this result to freeS-acts of infinite rank.Research supported by Natural Sciences and Engineering Research Council of Canada Research Grant A4494     

Semigroups embeddable in hyperplane face monoids

The left regular band structure on a hyperplane arrangement and its representation theory provide an important connection between semigroup theory and algebraic combinatorics. A finite semigroup embeds in a real hyperplane face monoid if and only if it is in the quasivariety generated by the monoid obtained by adjoining an identity to the two-element left zero semigroup. We prove that this quasivariety is on the one hand polynomial time decidable, and on the other minimally non-finitely based. A similar result is obtained for the semigroups embeddable in complex hyperplane semigroups.     

Endomorphism monoids and topological subgraphs of graphs

We prove, that, given a finite graph Y there exists a finite monoid (semigroup with unity) M such that any graph X whose endomorphism monoid is isomorphic to M contains a subdivision of Y. This contrasts with several known results on the simultaneous prescribability of the endomorphism monoid and various graph theoretical properties of a graph. It is also related to the analogous problems on graphs having a given permutation group as a restriction of their automorphism group to an invariant subset.     

On a Class of Lattice Ordered Inverse Semigroups

It is well known that the free group on a non-empty set can be totally ordered and, further, that each compatible latttice ordering on a free group is a total ordering. On the other hand, Saitô has shown that no non-trivial free inverse semigroup can be totally ordered. In this note we show, however, that every free inverse monoid admits compatible lattice orderings which are closely related to the total orderings on free groups.These orderings are natural in the sense that the imposed partial ordering on the idempotents coincides with the natural partial ordering. For this to happen in a lattice ordered inverse semigroup, the idempotents must form a distributive lattice. The method of construction of the lattice orderings on free inverse monoids can be applied to show that naturally lattice ordered inverse semigroups with a given distributive lattice E of idempotents can have arbitrary Green's relation structure. Analogous results hold for naturally -semilatticed inverse semigroups. In this case, there is no restriction on the semilattice E of idempotents.We also show that every compatible lattice ordering on the free monogenic inverse monoid is of the type considered here. This permits us to prove that there are precisely eight distinct compatible lattice orderings on this semigroup. They belong to two families, each of which contains four members, of conjuguate lattice orderings.     

The identities of the free product of two trivial semigroups

We exhibit an example of a finitely presented semigroup S with a minimum number of relations such that the identities of S have a finite basis while the monoid obtained by adjoining 1 to S admits no finite basis for its identities. Our example is the free product of two trivial semigroups.     

Free idempotent generated semigroups: subsemigroups,retracts and maximal subgroups

Let S be a subsemigroup of a semigroup T and let IG(E) and IG(F) be the free idempotent generated semigroups over the biordered sets of idempotents of E of S and F of T, respectively. We examine the relationship between IG(E) and IG(F), including the case where S is a retract of T. We give su?cient conditions satisfied by T and S such that for any e∈E, the maximal subgroup of IG(E) with identity e is isomorphic to the corresponding maximal subgroup of IG(F). We then apply this result to some special cases and, in particular, to that of the partial endomorphism monoid PEnd A and the endomorphism monoid EndA of an independence algebra A of finite rank. As a corollary, we obtain Dolinka’s reduction result for the case where A is a finite set.     

On finitely related semigroups

An algebraic structure is finitely related (has finite degree) if its term functions are determined by some finite set of finitary relations. We show that the following finite semigroups are finitely related: commutative semigroups, 3-nilpotent monoids, regular bands, semigroups with a single idempotent, and Clifford semigroups. Further we provide the first example of a semigroup that is not finitely related: the 6-element Brandt monoid. This answers a question by Davey, Jackson, Pitkethly, and Szabó from Davey et al. (Semigroup Forum, 83(1):89–122, 2011).