scholarly journals On regularity preservation in a semigroup

2004 ◽  
Vol 69 (1) ◽  
pp. 69-86 ◽  
Author(s):  
J.B. Hickey
Keyword(s):  
Regular Semigroup ◽  
The Other ◽  
Maximal Elements ◽  
Minimal Elements ◽  
Completely Simple Semigroups ◽  
Completely Simple ◽  
Natural Way

We consider certain subsets of a semigroup S, defined mainly by conditions involving regularity preservation. In particular, the regular base B(S) of S may be regarded as a generalisation of the zero ideal in a semigroup with zero; if it non-empty then S is E-inversive. The other subsets considered are related in a natural way either to B(S) or to the set RP(S) of regularity-preserving elements in S. In a regular semigroup (equipped with the Hartwig-Nambooripad order) each of these subsets contains either minimal elements only or maximal elements only. The relationships between the subsets are discussed, and some characterisations of completely simple semigroups are obtained.

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.

Completely simple semigroups: free products, free semigroups and varieties

1981 ◽  
Vol 88 (3-4) ◽  
pp. 293-313 ◽  
Author(s):  
P. R. Jones
Keyword(s):  
Matrix Representation ◽  
Maximal Subgroups ◽  
Lattice Of Varieties ◽  
Free Products ◽  
Completely Simple Semigroups ◽  
Countable Rank ◽  
Free Semigroups ◽  
Abelian Subgroups ◽  
And Inversion ◽  
Completely Simple

SynopsisThe class CS of completely simple semigroups forms a variety under the operations of multiplication and inversion (x−1 being the inverse of x in its ℋ-class). We determine a Rees matrix representation of the CS-free product of an arbitrary family of completely simple semigroups and deduce a description of the free completely simple semigroups, whose existence was proved by McAlister in 1968 and whose structure was first given by Clifford in 1979. From this a description of the lattice of varieties of completely simple semigroups is given in terms of certain subgroups of a free group of countable rank. Whilst not providing a “list” of identities on completely simple semigroups it does enable us to deduce, for instance, the description of all varieties of completely simple semigroups with abelian subgroups given by Rasin in 1979. It also enables us to describe the maximal subgroups of the “free” idempotent-generated completely simple semigroups T(α, β) denned by Eberhart et al. in 1973 and to show in general the maximal subgroups of the “V-free” semigroups of this type (which we define) need not be free in any variety of groups.

3. Formation of bilateral contracts

2018 ◽  
Author(s):  
Paul S. Davies
Keyword(s):  
The Other ◽  
Bilateral Contracts ◽  
Bilateral Contract ◽  
Natural Way

This chapter analyses the key elements traditionally required for the formation of a bilateral contract. Contracts are bargains. The natural way to make a bargain is for one side to propose the terms and the other to agree to them. So contracts are almost invariably made by a process of offer and acceptance. However, the lack of offer and acceptance does not necessarily preclude the existence of a contract, if a bargain can be discerned from the facts in some other way. The chapter begins by explaining what constitutes an offer, and discusses various common scenarios. It then examines the requirements of acceptance, since this is what is required for a contract to be concluded. It considers the possibilities that an offer might be revoked by the offeror; or rejected by the offeree; or the offeree might ask for further information; or the offer might lapse.

The Word Problem for Orthogroups

1981 ◽  
Vol 33 (4) ◽  
pp. 893-900 ◽  
Author(s):  
J. A. Gerhard ◽  
Mario Petrich
Keyword(s):  
Regular Semigroup ◽  
Maximal Subgroup ◽  
Word Problem ◽  
Unary Operation ◽  
Group Inverse ◽  
Completely Regular Semigroup ◽  
Completely Regular ◽  
Image Position ◽  
Natural Way

A semigroup which is a union of groups is said to be completely regular. If in addition the idempotents form a subsemigroup, the semigroup is said to be orthodox and is called an orthogroup. A completely regular semigroup S is provided in a natural way with a unary operation of inverse by letting a-l for a ∈ S be the group inverse of a in the maximal subgroup of S to which a belongs. This unary operation satisfies the identities(1)(2)(3)In fact a completely regular semigroup can be defined as a unary semigroup (a semigroup with an added unary operation) satisfying these identities. An orthogroup can be characterized as a completely regular semigroup satisfying the additional identity(4)

scholarly journals CHARACTERIZING SOME COMPLETELY REGULAR SEMIGROUPS BY THEIR SUBSEMIGROUPS

2013 ◽  
Vol 94 (3) ◽  
pp. 397-416 ◽  
Author(s):  
MARIO PETRICH
Keyword(s):  
Regular Semigroups ◽  
Completely Regular ◽  
Completely Simple Semigroups ◽  
Completely Regular Semigroups ◽  
Completely Simple

AbstractWe consider several familiar varieties of completely regular semigroups such as groups and completely simple semigroups. For each of them, we characterize their members in terms of absence of certain kinds of subsemigroups, as well as absence of certain divisors, and in terms of a homomorphism of a concrete semigroup into the semigroup itself. For each of these varieties $ \mathcal{V} $ we determine minimal non-$ \mathcal{V} $ varieties, provide a basis for their identities, determine their join and give a basis for its identities. Most of this is complete; one of the items missing is a basis for identities for minimal nonlocal orthogroups. Three tables and a figure illustrate the results obtained.

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