A Subdirect Product and Several Sufficient Conditions about a Strong Semilattice of Inverse Semigroups

Subdirect Products ◽

Bandelt and Petrich (1982) proved that an inversive semiringSis a subdirect product of a distributive lattice and a ring if and only ifSsatisfies certain conditions. The aim of this paper is to obtain a generalized version of this result. The main purpose of this paper however, is to investigate, what new necessary and sufficient conditions need we impose on an inversive semiring, so that, in its aforesaid representation as a subdirect product, the ring involved can be gradually enriched to a field. Finally, we provide a construction of fullE-inversive semirings, which are subdirect products of a semilattice and a ring.

A simplification of D'Alarcao's idempotent separating extensions of inverse semigroups

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171278000393 ◽

1978 ◽

Vol 1 (3) ◽

pp. 393-396

Author(s):

Constance C. Edwards

Keyword(s):

Inverse Semigroup ◽

Sufficient Conditions ◽

Inverse Semigroups ◽

In [2] D'Alarcao states necessary and sufficient conditions for the attainment of an idempotent-separating extension of an inverse semigroup. To do this D'Alarcao needed essentially three mappings satisfying thirteen conditions. In this paper we show that one can achieve the same results with two mappings satisfying eight conditions.

Some completely semisimple HNN-extensions of inverse semigroups

10.1142/s0218196719500711 ◽

2019 ◽

Vol 30 (02) ◽

pp. 217-243

Author(s):

Mohammed Abu Ayyash ◽

Alessandra Cherubini

Keyword(s):

Inverse Semigroup ◽

Sufficient Conditions ◽

Inverse Semigroups ◽

Bicyclic Semigroup ◽

Hnn Extensions ◽

Completely Semisimple

We give necessary and sufficient conditions in order that lower bounded HNN-extensions of inverse semigroups and HNN-extensions of finite inverse semigroups are completely semisimple semigroups. Since it is well known that an inverse semigroup is completely semisimple if and only if it does not contain a copy of the bicyclic semigroup, we first characterize such HNN-extensions containing a bicyclic subsemigroup making use of the special feature of their Schützenberger automata.

BICYCLIC SUBSEMIGROUPS IN AMALGAMS OF FINITE INVERSE SEMIGROUPS

10.1142/s021819671000556x ◽

2010 ◽

Vol 20 (01) ◽

pp. 89-113 ◽

Cited By ~ 7

Author(s):

EMANUELE RODARO

Keyword(s):

Inverse Semigroup ◽

Free Product ◽

Sufficient Conditions ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Inverse Semigroups ◽

Amalgamated Free Product ◽

Free Product With Amalgamation ◽

Completely Semisimple

It is well known that an inverse semigroup is completely semisimple if and only if it does not contain a copy of the bicyclic semigroup. We characterize the amalgams [S1, S2; U] of two finite inverse semigroups S1, S2whose free product with amalgamation is completely semisimple and we show that checking whether the amalgamated free product of finite inverse semigroups contains a bicyclic subsemigroup is decidable by means of a polynomial time algorithm with respect to max {|S1|,|S2|}. Moreover we consider amalgams of finite inverse semigroups respecting the [Formula: see text]-order proving that the free product with amalgamation is completely semisimple and we also provide necessary and sufficient conditions for the [Formula: see text]-classes to be finite.

Two Remarks to Bifullness of Centers of Archimedean Atomic Lattice Effect Algebras

Acta Polytechnica ◽

10.14311/1398 ◽

2011 ◽

Vol 51 (4) ◽

Author(s):

M. Kalina

Keyword(s):

Effect Algebra ◽

Subdirect Product ◽

Sufficient Conditions ◽

Effect Algebras ◽

Lattice Effect Algebra ◽

Orthomodular Lattices ◽

Atomic Lattice ◽

Lattice Effect ◽

Archimedean Lattice ◽

Mv Algebras

Lattice effect algebras generalize orthomodular lattices as well as MV-algebras. This means that within lattice effect algebras it is possible to model such effects as unsharpness (fuzziness) and/or non-compatibility. The main problem is the existence of a state. There are lattice effect algebras with no state. For this reason we need some conditions that simplify checking the existence of a state. If we know that the center C(E) of an atomic Archimedean lattice effect algebra E (which is again atomic) is a bifull sublattice of E, then we are able to represent E as a subdirect product of lattice effect algebras Ei where the top element of each one of Ei is an atom of C(E). In this case it is enough if we find a state at least in one of Ei and we are able to extend this state to the whole lattice effect algebra E. In [8] an atomic lattice effect algebra E (in fact, an atomic orthomodular lattice) with atomic center C(E) was constructed, where C(E) is not a bifull sublattice of E. In this paper we show that for atomic lattice effect algebras E (atomic orthomodular lattices) neither completeness (and atomicity) of C(E) nor σ-completeness of E are sufficient conditions for C(E) to be a bifull sublattice of E.

On the Structure of Inverse Semigroup Amalgams

10.1142/s0218196797000265 ◽

1997 ◽

Vol 07 (05) ◽

pp. 577-604 ◽

Cited By ~ 10

Author(s):

Paul Bennett

Keyword(s):

Inverse Semigroup ◽

Free Product ◽

Sufficient Conditions ◽

Inverse Semigroups ◽

Maximal Subgroups ◽

Regular Semigroups ◽

Endomorphism Monoids ◽

Amalgamated Free Product ◽

This paper is the second of two papers devoted to the study of amalgamated free products of inverse semigroups. We use the characterization of the Schützenberger automata given previously by the author to obtain structural results and preservational properties of lower bounded amalgams. Haataja, Margolis and Meakin have shown that if [S1,S2;U is an amalgam of regular semigroups in which S1∩ S2=U is a full regular subsemigroup of S1 and S2, then the maximal subgroups of the amalgamated free product S1*U S2 may be described by the fundamental groups of certain bipartite graphs of groups. In this paper we show that the maximal subgroups of a lower bounded amalgam [S1,S2;U] are either isomorphic copies of subgroups of S1 and S2 or can be described by the same Bass-Serre theory characterization. It follows, as for the regular case, that if S1 and S2 are combinatorial, then the maximal subgroups of S1*U S2 are free. By studying the endomorphism monoids of the Schützenberger graphs we obtain a number of results concerning when inverse semigroup properties are preserved under the amalgamated free product construction. For example, necessary and sufficient conditions are given for S1*U S2 to be completely semisimple. Under a mild assumption we establish necessary and sufficient conditions for S1*U S2 to have finite ℛ-classes. This enables us to reprove a result of Cherubini, Meakin and Piochi on amalgams of free inverse semigroups. Finally we give sufficient conditions for S1*U S2 to be E-unitary.

Trace functions on inverse semigroup algebras

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700014854 ◽

1995 ◽

Vol 52 (3) ◽

pp. 359-372 ◽

Cited By ~ 13

Author(s):

D. Easdown ◽

W.D. Munn

Keyword(s):

Inverse Semigroup ◽

Sufficient Conditions ◽

Semigroup Algebra ◽

Inverse Semigroups ◽

Trace Function ◽

Complex Field ◽

Semigroup Algebras ◽

Complex Conjugation ◽

Completely Semisimple

Let S be an inverse semigroup and let F be a subring of the complex field containing 1 and closed under complex conjugation. This paper concerns the existence of trace functions on F[S], the semigroup algebra of S over F. Necessary and sufficient conditions on S are found for the existence of a trace function on F[S] that takes positive integral values on the idempotents of S. Although F[S] does not always admit a trace function, a weaker form of linear functional is shown to exist for all choices of S. This is used to show that the natural involution on F[S] is special. It also leads to the construction of a trace function on F[S] for the case in which F is the real or complex field and S is completely semisimple of a type that includes countable free inverse semigroups.

On a complete topological inverse polycyclic monoid

Carpathian Mathematical Publications ◽

10.15330/cmp.8.2.183-194 ◽

2016 ◽

Vol 8 (2) ◽

pp. 183-194 ◽

Cited By ~ 4

Author(s):

S.O. Bardyla ◽

O.V. Gutik

Keyword(s):

Sufficient Conditions ◽

Inverse Semigroups ◽

Infinite Cardinal ◽

Inverse Monoid ◽

Inverse Topology

We give sufficient conditions when a topological inverse $\lambda$-polycyclic monoid $P_{\lambda}$ is absolutely $H$-closed in the class of topological inverse semigroups. For every infinite cardinal $\lambda$ we construct the coarsest semigroup inverse topology $\tau_{mi}$ on $P_\lambda$ and give an example of a topological inverse monoid $S$ which contains the polycyclic monoid $P_2$ as a dense discrete subsemigroup.

On the structure of symmetric n-ary bands

10.1142/s0218196721500478 ◽

2021 ◽

pp. 1-20

Author(s):

Jimmy Devillet ◽

Pierre Mathonet

Keyword(s):

Structure Theorem ◽

Abelian Groups ◽

Sufficient Conditions ◽

Semilattice Decomposition ◽

Strong Semilattice ◽

We study the class of symmetric [Formula: see text]-ary bands. These are [Formula: see text]-ary semigroups [Formula: see text] such that [Formula: see text] is invariant under the action of permutations and idempotent, i.e., satisfies [Formula: see text] for all [Formula: see text]. We first provide a structure theorem for these symmetric [Formula: see text]-ary bands that extends the classical (strong) semilattice decomposition of certain classes of bands. We introduce the concept of strong [Formula: see text]-ary semilattice of [Formula: see text]-ary semigroups and we show that the symmetric [Formula: see text]-ary bands are exactly the strong [Formula: see text]-ary semilattices of [Formula: see text]-ary extensions of Abelian groups whose exponents divide [Formula: see text]. Finally, we use the structure theorem to obtain necessary and sufficient conditions for a symmetric [Formula: see text]-ary band to be reducible to a semigroup.

Semigroups whose right ideals are finitely generated

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091521000638 ◽

2021 ◽

pp. 1-36

Author(s):

Craig Miller

Keyword(s):

Sufficient Conditions ◽

Chain Condition ◽

Finitely Generated ◽

Equivalent Formulation ◽

Regular Semigroups ◽

Strong Semilattice ◽

Ascending Chain Condition ◽

Completely Simple

Abstract We call a semigroup $S$ weakly right noetherian if every right ideal of $S$ is finitely generated; equivalently, $S$ satisfies the ascending chain condition on right ideals. We provide an equivalent formulation of the property of being weakly right noetherian in terms of principal right ideals, and we also characterize weakly right noetherian monoids in terms of their acts. We investigate the behaviour of the property of being weakly right noetherian under quotients, subsemigroups and various semigroup-theoretic constructions. In particular, we find necessary and sufficient conditions for the direct product of two semigroups to be weakly right noetherian. We characterize weakly right noetherian regular semigroups in terms of their idempotents. We also find necessary and sufficient conditions for a strong semilattice of completely simple semigroups to be weakly right noetherian. Finally, we prove that a commutative semigroup $S$ with finitely many archimedean components is weakly (right) noetherian if and only if $S/\mathcal {H}$ is finitely generated.