JOINS OF INVERSE SEMIGROUP VARIETIES AND BAND VARIETIES

1991 ◽  
Vol 01 (03) ◽  
pp. 371-385 ◽  
Author(s):  
PETER R. JONES ◽  
PETER G. TROTTER
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Lattice Of Varieties ◽  
Orthodox Semigroups ◽  
Semigroup Varieties

The joins in the title are considered within two contexts: (I) the lattice of varieties of regular unary semigroups, and (II) the lattice of e-varieties (or bivarieties) of orthodox semigroups. It is shown that in each case the set of all such joins forms a proper sublattice of the respective join of the variety I of all inverse semigroups and the variety B of all bands; each member V of this sublattice is determined by V ∩ I and V ∩ B. All subvarieties of the join of I with the variety RB of regular bands are so determined. However, there exist uncountably many subvarieties (or sub-bivarieties) of the join I ∨ B, all of which contain I and all of whose bands are regular.

scholarly journals Semigroups whose idempotents form a subsemigroup

1990 ◽  
Vol 41 (2) ◽  
pp. 161-184 ◽  
Author(s):  
Jean-Camille Birget ◽  
Stuart Margolis ◽  
John Rhodes
Keyword(s):  
Inverse Semigroup ◽  
Finite Semigroup ◽  
Orthodox Semigroup ◽  
Inverse Semigroups ◽  
Type Ii ◽  
Finite Semigroups ◽  
Case Type ◽  
Orthodox Semigroups

We prove that if the “type-II-construct” subsemigroup of a finite semigroup S is regular, then the “type-II” subsemigroup of S is computable (actually in this case, type-II and type-II-construct are equal). This, together with certain older results about pseudo-varieties of finite semigroups, leads to further results:(1) We get a new proof of Ash's theorem: If the idempotents in a finite semigroup S commute, then S divides a finite inverse semigroup. Equivalently: The pseudo-variety generated by the finite inverse semigroups consists of those finite semigroups whose idempotents commute.(2) We prove: If the idempotents of a finite semigroup S form a subsemigroup then S divides a finite orthodox semigroup. Equivalently: The pseudo-variety generated by the finite orthodox semigroups consists of those finite semigroups whose idempotents form a subsemigroup.(3) We prove: The union of all the subgroups of a semigroup S forms a subsemigroup if and only if 5 belongs to the pseudo-variety u * G if and only if Sn belongs to u. Here u denotes the pseudo-variety of finite semigroups which are unions of groups.For these three classes of semigroups, type-II is equal to type-II construct.

EXTENSIONS OF REGULAR ORTHOGROUPS BY INVERSE SEMIGROUPS

1995 ◽  
Vol 05 (03) ◽  
pp. 317-342 ◽  
Author(s):  
BERND BILLHARDT
Keyword(s):  
Inverse Semigroup ◽  
Recent Result ◽  
Semidirect Product ◽  
Orthodox Semigroup ◽  
Inverse Semigroups ◽  
Normal Extension ◽  
Group Congruence ◽  
Completely Regular ◽  
Orthodox Semigroups

Let V be a variety of regular orthogroups, i.e. completely regular orthodox semigroups whose band of idempotents is regular. Let S be an orthodox semigroup which is a (normal) extension of an orthogroup K from V by an inverse semigroup G, that is, there is a congruence ρ on S such that the semigroup ker ρ of all idempotent related elements of S is isomorphic to K and S/ρ≅G. It is shown that S can be embedded into an orthodox subsemigroup T of a double semidirect product A**G where A belongs to V. Moreover T itself can be chosen to be an extension of a member from V by G. If in addition ρ is a group congruence we obtain a recent result due to M.B. Szendrei [16] which says that each orthodox semigroup which is an extension of a regular orthogroup K by a group G can be embedded into a semidirect product of an orthogroup K′ by G where K′ belongs to the variety of orthogroups generated by K.

scholarly journals A NOTE ON THE HOWSON PROPERTY IN INVERSE SEMIGROUPS

2016 ◽  
Vol 94 (3) ◽  
pp. 457-463 ◽  
Author(s):  
PETER R. JONES
Keyword(s):  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finitely Generated ◽  
Necessary And Sufficient

An algebra has the Howson property if the intersection of any two finitely generated subalgebras is again finitely generated. A simple necessary and sufficient condition is given for the Howson property to hold on an inverse semigroup with finitely many idempotents. In addition, it is shown that any monogenic inverse semigroup has the Howson property.

scholarly journals Inverse semigroups and their natural order

1978 ◽  
Vol 19 (1) ◽  
pp. 59-65 ◽  
Author(s):  
H. Mitsch
Keyword(s):  
Inverse Semigroup ◽  
Partial Order ◽  
Homomorphic Image ◽  
Point Of View ◽  
Inverse Semigroups ◽  
Natural Order ◽  
Theoretical Point ◽  
Trivial Group

The natural order of an inverse semigroup defined by a ≤ b ⇔ a′b = a′a has turned out to be of great importance in describing the structure of it. In this paper an order-theoretical point of view is adopted to characterise inverse semigroups. A complete description is given according to the type of partial order an arbitrary inverse semigroup S can possibly admit: a least element of (S, ≤) is shown to be the zero of (S, ·); the existence of a greatest element is equivalent to the fact, that (S, ·) is a semilattice; (S, ≤) is directed downwards, if and only if S admits only the trivial group-homomorphic image; (S, ≤) is totally ordered, if and only if for all a, b ∈ S, either ab = ba = a or ab = ba = b; a finite inverse semigroup is a lattice, if and only if it admits a greatest element. Finally formulas concerning the inverse of a supremum or an infimum, if it exists, are derived, and right-distributivity and left-distributivity of multiplication with respect to union and intersection are shown to be equivalent.

scholarly journals FACTORIZATION THEOREMS FOR MORPHISMS OF ORDERED GROUPOIDS AND INVERSE SEMIGROUPS

2001 ◽  
Vol 44 (3) ◽  
pp. 549-569 ◽  
Author(s):  
Benjamin Steinberg
Keyword(s):  
Inverse Semigroup ◽  
Regular Semigroup ◽  
Classical Theory ◽  
Derived Category ◽  
Inverse Semigroups ◽  
Subject Classification

AbstractAdapting the theory of the derived category to ordered groupoids, we prove that every ordered functor (and thus every inverse and regular semigroup homomorphism) factors as an enlargement followed by an ordered fibration. As an application, we obtain Lawson’s version of Ehresmann’s Maximum Enlargement Theorem, from which can be deduced the classical theory of idempotent-pure inverse semigroup homomorphisms and $E$-unitary inverse semigroups.AMS 2000 Mathematics subject classification: Primary 20M18; 20L05; 20M17

scholarly journals Embedding inverse semigroups of homeomorphisms on closed subsets

1977 ◽  
Vol 18 (2) ◽  
pp. 199-207 ◽  
Author(s):  
Bridget Bos Baird
Keyword(s):  
Topological Space ◽  
Inverse Semigroup ◽  
Inverse Semigroups ◽  
Topological Spaces ◽  
Locally Compact ◽  
Compact Metric Space ◽  
Countable Space ◽  
The One ◽  
One Point Compactification ◽  
First Countable Space

All topological spaces here are assumed to be T2. The collection F(Y)of all homeomorphisms whose domains and ranges are closed subsets of a topological space Y is an inverse semigroup under the operation of composition. We are interested in the general problem of getting some information about the subsemigroups of F(Y) whenever Y is a compact metric space. Here, we specifically look at the problem of determining those spaces X with the property that F(X) is isomorphic to a subsemigroup of F(Y). The main result states that if X is any first countable space with an uncountable number of points, then the semigroup F(X) can be embedded into the semigroup F(Y) if and only if either X is compact and Y contains a copy of X, or X is noncompact and locally compact and Y contains a copy of the one-point compactification of X.

Inverse semigroups generated by a pair of subgroups

1977 ◽  
Vol 77 (1-2) ◽  
pp. 9-22 ◽  
Author(s):  
D. B. McAlister
Keyword(s):  
Inverse Semigroup ◽  
Free Product ◽  
Homomorphic Image ◽  
Main Part ◽  
Structure Theorem ◽  
Inverse Semigroups ◽  
Infinite Cyclic Group ◽  
The Third ◽  
Group A ◽  
Maximum Group

SynopsisThe aim of this paper is to describe the free product of a pair G, H of groups in the category of inverse semigroups. Since any inverse semigroup generated by G and H is a homomorphic image of this semigroup, this paper can be regarded as asking how large a subcategory, of the category of inverse semigroups, is the category of groups? In this light, we show that every countable inverse semigroup is a homomorphic image of an inverse subsemigroup of the free product of two copies of the infinite cyclic group. A similar result can be obtained for arbitrary cardinalities. Hence, the category of inverse semigroups is generated, using algebraic constructions by the subcategory of groups.The main part of the paper is concerned with obtaining the structure of the free product G inv H, of two groups G, H in the category of inverse semigroups. It is shown in section 1 that G inv H is E-unitary; thus G inv H can be described in terms of its maximum group homomorphic image G gp H, the free product of G and H in the category of groups, and its semilattice of idempotents. The second section considers some properties of the semilattice of idempotents while the third applies these to obtain a representation of G inv H which is faithful except when one group is a non-trivial finite group and the other is trivial. This representation is used in section 4 to give a structure theorem for G inv H. In this section, too, the result described in the first paragraph is proved. The last section, section 5, consists of examples.

REPRESENTATIONS OF LOCALLY INVERSE *-SEMIGROUPS

1996 ◽  
Vol 06 (05) ◽  
pp. 541-551
Author(s):  
TERUO IMAOKA ◽  
ISAMU INATA ◽  
HIROAKI YOKOYAMA
Keyword(s):  
Inverse Semigroup ◽  
Wreath Product ◽  
Representation Theorem ◽  
Generalized Inverse ◽  
Inverse Semigroups ◽  
Locally Inverse Semigroup ◽  
The One ◽  
Locally Inverse Semigroups ◽  
Generalized Inverse Semigroups

The first author obtained a generalization of Preston-Vagner Representation Theorem for generalized inverse *-semigroups. In this paper, we shall generalize their results for locally inverse *-semigroups. Firstly, by introducing a concept of a π-set (which is slightly different from the one in [7]), we shall construct the π-symmetric locally inverse *-semigroup on a π-set, and show that any locally inverse *-semigroup can be embedded up to *-isomorphism in the π-symmetric locally inverse semigroup on a π-set. Moreover, we shall obtain that the wreath product of locally inverse *-semigroups is also a locally inverse *-semigroup.

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