The ϰ°-relation on the congruence lattice of a regular semigroup with an inverse transversal

2011 ◽  
Vol 28 (5) ◽  
pp. 1061-1074
Author(s):  
Ying Ying Feng ◽  
Li Min Wang
Keyword(s):  
Regular Semigroup ◽  
Congruence Lattice ◽  
Inverse Transversal

scholarly journals On quasi-orthodox semigroups with inverse transversals

1997 ◽  
Vol 40 (3) ◽  
pp. 505-514 ◽  
Author(s):  
T. S. Blyth ◽  
M. H. Almeida Santos
Keyword(s):  
Regular Semigroup ◽  
Inverse Transversal ◽  
Inverse Subsemigroup ◽  
Orthodox Semigroups

An inverse transversal of a regular semigroup S is an inverse subsemigroup So that contains precisely one inverse of each element of S. Here we consider the case where S is quasi-orthodox. We give natural characterisations of such semigroups and consider various properties of congruences.

scholarly journals A simplistic approach to inverse transversals

1996 ◽  
Vol 39 (1) ◽  
pp. 57-69 ◽  
Author(s):  
T. S. Blyth ◽  
M. H. Almeida Santos
Keyword(s):  
Regular Semigroup ◽  
Inverse Transversal ◽  
Inverse Subsemigroup

An inverse transversal of a regular semigroup S is an inverse subsemigroup that contains precisely one inverse of each element of S. In the literature there are three known types of inverse transversal, namely those that are multiplicative, those that are weakly multiplicative, and those that form quasi-ideals. Here, by considering natural ways in which certain words can be simplified, we reveal four new types of inverse transversal. All of these can be illustrated nicely in examples that are based on 2 × 2 matrices.

scholarly journals A special sublattice of the congruence lattice of a regular semigroup

1997 ◽  
Vol 40 (3) ◽  
pp. 457-472 ◽  
Author(s):  
Mario Petrich
Keyword(s):  
Distributive Lattice ◽  
Regular Semigroup ◽  
Congruence Lattice ◽  
Regular Semigroups ◽  
Generators And Relations ◽  
Completely Regular ◽  
Completely Simple Semigroups ◽  
Special Cases ◽  
Completely Regular Semigroups ◽  
Completely Simple

Let S be a regular semigroup and be its congruence lattice. For ρ ∈ , we consider the sublattice Lρ of generated by the congruences pw where w ∈ {K, k, T, t}* and w has no subword of the form KT, TK, kt, tk. Here K, k, T, t are the operators on induced by the kernel and the trace relations on . We find explicitly the least lattice L whose homomorphic image is Lρ for all ρ ∈ and represent it as a distributive lattice in terms of generators and relations. We also consider special cases: bands of groups, E-unitary regular semigroups, completely simple semigroups, rectangular groups as well as varieties of completely regular semigroups.

REPRESENTATIONS OF SEMIGROUPS BY TRANSFORMATIONS AND THE CONGRUENCE LATTICE OF AN EVENTUALLY REGULAR SEMIGROUP

1996 ◽  
Vol 06 (06) ◽  
pp. 655-685 ◽  
Author(s):  
K. AUINGER ◽  
T.E. HALL
Keyword(s):  
Regular Semigroup ◽  
Congruence Lattice ◽  
Left Zero ◽  
Completely Simple Semigroups ◽  
Arbitrary Semigroup ◽  
Eventually Regular Semigroup ◽  
Completely Simple

On any eventually regular semigroup S, congruences ν, μL, μR, μ, K, KL, KR, ζ are introduced which are the greatest congruences over: nil-extensions (n.e.) of completely simple semigroups, n.e. of left groups, n.e. of right groups, n.e. of groups, n.e. of rectangular bands, n.e. of left zero semigroups, n.e. of right zero semigroups, nil-semigroups, respectively. Each of these congruences is induced by a certain representation of S which is defined on an arbitrary semigroup. These congruences play an important role in the study of lattices of varieties, pseudovarieties and existence varieties. The investigation also leads to eight complete congruences U, Tt, Tr, T, K, Kl, Kr, Z on the congruence lattice Con (S) of S.

Regular semigroups with a multiplicative inverse transversal

1982 ◽  
Vol 92 (3-4) ◽  
pp. 253-270 ◽  
Author(s):  
T. S. Blyth ◽  
R. McFadden
Keyword(s):  
Regular Semigroup ◽  
Inverse Transversal ◽  
Multiplicative Inverse ◽  
Multiplicative Property ◽  
Regular Semigroups ◽  
Inverse Subsemigroup

SynopsisBy an inverse transversal of a regular semigroup S we mean an inverse subsemigroup that contains a single inverse of every element of S. A certain multiplicative property (which in the case of a band is equivalent to normality) is imposed on an inverse transversal and a complete description of the structure of S is obtained.

scholarly journals The kernel relation for a strict extension of certain regular semigroups

1996 ◽  
Vol 38 (3) ◽  
pp. 347-357 ◽  
Author(s):  
Mario Petrich
Keyword(s):  
Regular Semigroup ◽  
Congruence Lattice ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Regular Semigroups ◽  
Strict Extension ◽  
Necessary And Sufficient ◽  
Special Classes

Let R be a regular semigroup and denote by (R) its congruence lattice. For , the kernel of pis the set ker . The relation K on (R) defined by λKp if ker λ = ker p is the kernel relation on (R). In general, K is a complete ∩-congruence but it is not a v-congruence. In view of the importance of the kernel-trace approach to the study of congruences on a regular semigroup (the trace of p is its restriction to idempotents of R), it is of considerable interest to determine necessary and sufficient conditions on R in order for K to be a congruence. This being in general a difficult task, one restricts attention to special classes of regular semigroups. For a background on this subject, consult [1].

scholarly journals Decompositions of the congruence lattice of a semigroup

1980 ◽  
Vol 23 (2) ◽  
pp. 193-198 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Regular Semigroup ◽  
Congruence Lattice ◽  
Alternative Proof

The purpose of this note is to extend the results of Reilly and Scheiblich (6) (see also Scheiblich (7) and Hall (2)) on the θ-class decomposition of the congruence lattice of a regular semigroup and, at the same time, to provide an alternative proof of these results.

scholarly journals On weakly multiplicative inverse transversals

1994 ◽  
Vol 37 (1) ◽  
pp. 91-99 ◽  
Author(s):  
T. S. Blyth ◽  
M. H. Almeida Santos
Keyword(s):  
Regular Semigroup ◽  
Inverse Transversal ◽  
Multiplicative Inverse

We show that an inverse transversal of a regular semigroup is multiplicative if and only if it is both weakly multiplicative and a quasi-ideal. Examples of quasi-ideal inverse transversals that are not multiplicative are known. Here we give an example of a weakly multiplicative inverse transversal that is not multiplicative. An interesting feature of this example is that it also serves to show that, in an ordered regular semigroup in which every element x has a biggest inverse x0, the mapping x↦x00 is not in general a closure; nor is x↦x** in a principally ordered regular semigroup.

scholarly journals Construction of regular semigroups with inverse transversals

1989 ◽  
Vol 32 (1) ◽  
pp. 41-51 ◽  
Author(s):  
Tatsuhiko Saito
Keyword(s):  
Regular Semigroup ◽  
Unique Element ◽  
Inverse Transversal ◽  
Regular Semigroups ◽  
Inverse Subsemigroup

Let S be a regular semigroup. An inverse subsemigroup S° of S is an inverse transversal if |V(x)∩S°| = 1 for each x∈S, where V(x) denotes the set of inverses of x. In this case, the unique element of V(x)∩S° is denoted by x°, and x°° denotes (x°)–1. Throughout this paper S denotes a regular semigroup with an inverse transversal S°, and E(S°) = E° denotes the semilattice of idempotents of S°. The sets {e∈S:ee° = e} and {f∈S:f°f=f} are denoted by Is and Λs, respectively, or simply I and Λ. Though each element of these sets is idempotent, they are not necessarily sub-bands of S. When both I and Λ are sub-bands of S, S° is called an S-inverse transversal. An inverse transversal S° is multiplicative if x°xyy°∈E°, and S° is weakly multiplicative if (x°xyy°)°∈E° for every x, y∈S. A band B is left [resp. right] regular if e f e = e f [resp. e f e = f e], and B is left [resp. right] normal if e f g = e g f [resp. e f g = f e g] for every e, f, g∈B. A subset Q of S is a quasi-ideal of S if QSQ ⊆ S.

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