scholarly journals On existence varieties of orthodox semigroups

1995 ◽  
Vol 58 (1) ◽  
pp. 100-125 ◽  
Author(s):  
J. Doyle
Keyword(s):  
Inverse Semigroups ◽  
Direct Products ◽  
Regular Semigroups ◽  
Existence Variety ◽  
Orthodox Semigroups

AbstractAn existence variety of regular semigroups is a class of regular semigroups which is closed under the operations of forming all homomorphic images, all regular subsemigroups and all direct products. In this paper we generalize results on varieties of inverse semigroups to existence varieties of orthodox semigroups.

scholarly journals Identities for existence varieties of regular semigroups

1989 ◽  
Vol 40 (1) ◽  
pp. 59-77 ◽  
Author(s):  
T.E. Hall
Keyword(s):  
Type Theorem ◽  
Inverse Semigroups ◽  
Direct Products ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Existence Variety ◽  
Completely Regular Semigroups ◽  
Locally Inverse Semigroups ◽  
Orthodox Semigroups ◽  
Regular Rings

A natural concept of variety for regular semigroups is introduced: an existence variety (or e-variety) of regular semigroups is a class of regular semigroups closed under the operations H, Se, P of taking all homomorphic images, regular subsernigroups and direct products respectively. Examples include the class of orthodox semigroups, the class of (regular) locally inverse semigroups and the class of regular E-solid semigroups. The lattice of e-varieties of regular semigroups includes the lattices of varieties of inverse semigroups and of completely regular semigroups. A Birkhoff-type theorem is proved, showing that each e-variety is determined by a set of identities: such identities are then given for many e-varieties. The concept is meaningful in universal algebra, and as for regular semigroups could give interesting results for e-varieties of regular rings.

On existence varieties of locally inverse semigroups

1994 ◽  
Vol 115 (2) ◽  
pp. 197-217 ◽  
Author(s):  
K. Auinger ◽  
J. Doyle ◽  
P. R. Jones
Keyword(s):  
Deep Structure ◽  
Inverse Semigroups ◽  
Direct Products ◽  
Regular Semigroups ◽  
Completely Simple Semigroups ◽  
Locally Inverse Semigroup ◽  
Existence Variety ◽  
Locally Inverse Semigroups ◽  
Completely Simple

AbstractA locally inverse semigroup is a regular semigroup S with the property that eSe is inverse for each idempotent e of S. Motivated by natural examples such as inverse semigroups and completely simple semigroups, these semigroups have been the subject of deep structure-theoretic investigations. The class ℒ ℐ of locally inverse semigroups forms an existence variety (or e-variety): a class of regular semigroups closed under direct products, homomorphic images and regular subsemigroups. We consider the lattice ℒ(ℒℐ) of e-varieties of such semigroups. In particular we investigate the operations of taking meet and join with the e-variety CS of completely simple semigroups. An important consequence of our results is a determination of the join of CS with the e-variety of inverse semigroups – it comprises the E-solid locally inverse semigroups. It is shown, however, that not every e-variety of E-solid locally inverse semigroups is the join of completely simple and inverse e-varieties.

Boolean Congruence Lattices of Orthodox Semigroups

1991 ◽  
Vol 43 (2) ◽  
pp. 225-241 ◽  
Author(s):  
Karl Auinger
Keyword(s):  
Inverse Semigroups ◽  
Regular Semigroups ◽  
Commutative Semigroups ◽  
Completely Regular ◽  
Clifford Semigroups ◽  
Congruence Lattices ◽  
Completely Regular Semigroups ◽  
Orthodox Semigroups

The problem of characterizing the semigroups with Boolean congruence lattices has been solved for several classes of semigroups. Hamilton [9] and the author of this paper [1] studied the question for semilattices. Hamilton and Nordahl [10] considered commutative semigroups, Fountain and Lockley [7,8] solved the problem for Clifford semigroups and idempotent semigroups, in [1] the author generalized their results to completely regular semigroups. Finally, Zhitomirskiy [19] studied the question for inverse semigroups.

scholarly journals On generalized Ehresmann semigroups

2017 ◽  
Vol 15 (1) ◽  
pp. 1132-1147
Author(s):  
Shoufeng Wang
Keyword(s):  
Inverse Semigroups ◽  
Inverse Transversal ◽  
Regular Semigroups ◽  
Admissible Triple ◽  
Orthodox Semigroups ◽  
Ehresmann Semigroups

Abstract As a generalization of the class of inverse semigroups, the class of Ehresmann semigroups is introduced by Lawson and investigated by many authors extensively in the literature. In particular, Gomes and Gould construct a fundamental Ehresmann semigroup CE from a semilattice E which plays for Ehresmann semigroups the role that TE plays for inverse semigroups, where TE is the Munn semigroup of a semilattice E. From a varietal perspective, Ehresmann semigroups are derived from reduction of inverse semigroups. In this paper, from varietal perspective Ehresmann semigroups are extended to generalized Ehresmann semigroups derived instead from normal orthodox semigroups (i.e. regular semigroups whose idempotents form normal bands) with an inverse transversal. We present here a semigroup C(I,Λ,E∘) from an admissible triple (I, Λ, E∘) that plays for generalized Ehresmann semigroups the role that CE from a semilattice E plays for Ehresmann semigroups. More precisely, we show that a semigroup is a fundamental generalized Ehresmann semigroup whose admissible triple is isomorphic to (I, Λ, E∘) if and only if it is (2,1,1,1)-isomorphic to a quasi-full (2,1,1,1)-subalgebra of C(I,Λ,E∘). Our results generalize and enrich some results of Fountain, Gomes and Gould on weakly E-hedges semigroups and Ehresmann semigroups.

THE EXISTENCE OF E-FREE OBJECTS IN E-VARIETIES OF REGULAR SEMIGROUPS

1992 ◽  
Vol 02 (04) ◽  
pp. 471-484 ◽  
Author(s):  
Y.T. YEH
Keyword(s):  
Free Object ◽  
Regular Semigroups ◽  
Free Objects ◽  
Existence Variety ◽  
Orthodox Semigroups

An existence variety (or e-variety) of regular semigroups is a class of regular semigroups which is closed under [Formula: see text], and ℍ. This concept was introduced by T.E. Hall and independently for orthodox semigroups by J. Kadourek and M.B. Szendrei who called them bivarieties. In this paper we prove the existence of e-free objects in each e-variety of E-solid regular semigroups and in each e-variety of locally inverse regular semigroups. By contrast, we show that there is no e-free object in other e-varieties.

ON PSEUDO-EHRESMANN SEMIGROUPS

2018 ◽  
Vol 105 (2) ◽  
pp. 257-288 ◽  
Author(s):  
SHOUFENG WANG
Keyword(s):  
Inverse Semigroups ◽  
Inverse Transversal ◽  
Multiplicative Inverse ◽  
New Approach ◽  
Regular Semigroups ◽  
Common Framework ◽  
Orthodox Semigroups ◽  
Ehresmann Semigroups

As generalizations of inverse semigroups, Ehresmann semigroups are introduced by Lawson and investigated by many authors extensively in the literature. In particular, Lawson has proved that the category of Ehresmann semigroups and admissible morphisms is isomorphic to the category of Ehresmann categories and strongly ordered functors, which generalizes the well-known Ehresmann–Schein–Nambooripad (ESN) theorem for inverse semigroups. From a varietal perspective, Ehresmann semigroups are derived from reducts of inverse semigroups. In this paper, inspired by the approach of Jones [‘A common framework for restriction semigroups and regular $\ast$-semigroups’, J. Pure Appl. Algebra216 (2012), 618–632], Ehresmann semigroups are extended from a varietal perspective to pseudo-Ehresmann semigroups derived instead from reducts of regular semigroups with a multiplicative inverse transversal. Furthermore, motivated by the method used by Gould and Wang [‘Beyond orthodox semigroups’, J. Algebra368 (2012), 209–230], we introduce the notion of inductive pseudocategories over admissible quadruples by which pseudo-Ehresmann semigroups are described. More precisely, we show that the category of pseudo-Ehresmann semigroups and (2,1,1,1)-morphisms is isomorphic to the category of inductive pseudocategories over admissible quadruples and pseudofunctors. Our work not only generalizes the result of Lawson for Ehresmann semigroups but also produces a new approach to characterize regular semigroups with a multiplicative inverse transversal.

scholarly journals Bifree objects in e-varieties of strict orthodox semigroups and the lattice of strict orthodox *-semigroup varieties

1993 ◽  
Vol 35 (1) ◽  
pp. 25-37 ◽  
Author(s):  
Karl Auinger
Keyword(s):  
Type Theorem ◽  
Orthodox Semigroup ◽  
Prima Facie ◽  
Direct Products ◽  
Regular Semigroups ◽  
Completely Regular ◽  
Free Objects ◽  
Completely Regular Semigroups ◽  
Definition Of ◽  
Orthodox Semigroups

For regular semigroups, the appropriate analogue of the concept of a variety seems to be that of an e(xistence)-variety, developed by Hall [6,7,8]. A class V of regular semigroups is an e-variety if it is closed under taking direct products, regular subsemigroups and homomorphic images. For orthodox semigroups, this concept has been introduced under the term “bivariety” by Kaďourek and Szendrei [12]. Hall showed that the collection of all e-varieties of regular semigroups forms a complete lattice under inclusion. Further, he proved a Birkhoff-type theorem: each e-variety is determined by a set of identities. For e-varieties of orthodox semigroups a similar result has been proved by Kaďourek and Szendrei. At variance with the case of varieties, prima facie the free objects in general do not exist for e-varieties. For instance, there is no free regular or free orthodox semigroup. This seems to be true for most of the naturally appearing e-varieties (except for cases of e-varieties which coincide with varieties of unary semigroups such as the classes of all inverse and completely regular semigroups, respectively). This is true if the underlying concept of free objects is denned as usual. Kaďourek and Szendrei adopted the definition of a free object according to e-varieties of orthodox semigroups by taking into account generalized inverses in an appropriate way. They called such semigroups bifree objects. These semigroups satisfy the properties one intuitively expects from the “most general members” of a given class of semigroups. In particular, each semigroup in the given class is a homomorphic image of a bifree object, provided the bifree objects exist on sets of any cardinality. Concerning existence, Kaďourek and Szendrei were able to prove that in any class of orthodox semigroups which is closed under taking direct products and regular subsemigroups, all bifree objects exist and are unique up to isomorphism. Further, similar to the case of varieties, there is an order inverting bijection between the fully invariant congruences on the bifree orthodox semigroup on an infinite set and the e-varieties of orthodox semigroups. Recently, Y. T. Yeh [22] has shown that suitable analogues to free objects exist in an e-variety V of regular semigroups if and only if all members of V are either E-solid or locally inverse.

On a class of Dubreil-Jacotin regular semigroups and a construction of Yamada

1977 ◽  
Vol 77 (1-2) ◽  
pp. 145-150 ◽  
Author(s):  
T. S. Blyth
Keyword(s):  
Structure Theorem ◽  
Present Approach ◽  
Inverse Semigroups ◽  
Regular Semigroups ◽  
Structure Theorems

SynopsisIn the publication [2] we obtained some structure theorems for certain Dubreil-Jacotin regular semigroups. A crucial observation in the course of investigating these types of ordered regular semigroups was that the (ordered) band of idempotents was normal. This is characteristic of a class of semigroups studied by Yamada [5] and called generalised inverse semigroups. Here we specialise a construction of Yamada to obtain a structure theorem that complements those in [2], The important feature of the present approach is the part played by the greatest elements that exist in each of the components in the semilattice decompositions involved.

scholarly journals A NEW CONSTRUCTION FOR REGULAR SEMIGROUPS WITH QUASI-IDEAL ORTHODOX TRANSVERSALS

2009 ◽  
Vol 86 (2) ◽  
pp. 177-187 ◽  
Author(s):  
XIANGJUN KONG ◽  
XIANZHONG ZHAO
Keyword(s):  
Regular Semigroup ◽  
Structure Theorem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Green’S Relations ◽  
Regular Semigroups ◽  
New Construction ◽  
Necessary And Sufficient ◽  
Equivalent Conditions ◽  
Orthodox Semigroups

AbstractIn any regular semigroup with an orthodox transversal, we define two sets R and L using Green’s relations and give necessary and sufficient conditions for them to be subsemigroups. By using R and L, some equivalent conditions for an orthodox transversal to be a quasi-ideal are obtained. Finally, we give a structure theorem for regular semigroups with quasi-ideal orthodox transversals by two orthodox semigroups R and L.

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.

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