Varieties of idempotent semigroups

doi:10.1007/BF02218673

Journal Article•DOI•

Varieties of idempotent semigroups

01 May 1970-Algebra and Logic (Kluwer Academic Publishers-Plenum Publishers)-Vol. 9, Iss: 3, pp 153-164

About: This article is published in Algebra and Logic.The article was published on 1970-05-01. It has received 99 citations till now. The article focuses on the topics: Idempotence.

...read moreread less

Citations

PDF

Open Access

More filters

Book Chapter•DOI•

Completely Regular Semigroups

[...]

Norman R. Reilly¹•Institutions (1)

Simon Fraser University¹

01 Jan 1990

TL;DR: A semigroup S is completely regular if and only if it is a disjoint union of groups as mentioned in this paper, and the theory of such objects must necessarily be readily derivable from that for groups.

...read moreread less

Abstract: A semigroup S is completely regular if and only if it is a disjoint union of groups. This concept, so simple in its formulation, has intrigued investigators for over forty years. In the early days, the underlying proposition was that the theory of such objects must necessarily be readily derivable from that for groups. The early work of Rees and Clifford gave some support to this notion. However, the work of recent years, especially that on varieties, has shown that the study of completely regular semigroups requires its own ingenious arsenal of tools.

...read moreread less

339 citations

Journal Article•DOI•

Identities for existence varieties of regular semigroups

[...]

T. E. Hall¹•Institutions (1)

Monash University, Clayton campus¹

01 Aug 1989-Bulletin of The Australian Mathematical Society

TL;DR: In this article, an existence variety of regular semigroups is introduced, which is a class of regular semiigroups closed under the operations H, Se, P of taking all homomorphic images, regular subsernigroups and direct products respectively.

...read moreread less

Abstract: 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

...read moreread less

79 citations

Cites background from "Varieties of idempotent semigroups"

...,bn are chosen in all the ways that make equations (1) valid in A....
[...]
..., bn G A such that equations (1) hold, and define f^(a\, • • •, a m ) = bj , for j = 1, 2, ....
[...]
...We remark that (1) a similar proof shows that for every inverse unary operation on (S, •), (S, •, ') does not satisfy (xy) = x"y"; and (2) a similar example can be obtained by replacing T in the construction by the 0-direct union of two copies of B2 - M°((l); 2, 2; A)....
[...]
...Let S be the combinatorial completely 0-simple semigroup M?((1); 3, 2; P) where...
[...]

Journal Article•DOI•

The lattice of completely regular semigroup varieties

[...]

Francis Pastijn¹•Institutions (1)

Marquette University¹

01 Aug 1990-Journal of The Australian Mathematical Society

TL;DR: The class of completely regular semigroups (CR) is a semigroup which is a union of groups as discussed by the authors, and the class CR of CRs forms a variety of the semigroup.

...read moreread less

Abstract: A completely regular semigroup is a semigroup which is a union of groups. The class CR of completely regular semigroups forms a variety. On the lattice L (CR) of completely regular semigroup varieties we define two closure operations which induce complete congruences. The consideration of a third complete congruence on L (CR) yields a subdirect decomposition of L (CR). Using these results we show that L (CR) is arguesian. This confirms the (tacit) conjecture that L (CR) is modular.

...read moreread less

69 citations

Cites background from "Varieties of idempotent semigroups"

...From [1], [3], [4] or [14] we have that V^ is either one of the varieties LRB, LNB, SL, LZ or T....
[...]
...The modularity of several particular sublattices of £(CR) was established in [1], [3], [4], [5], [7], [17], [18]....
[...]
...Using the above and the results of [1], [3] or [4] we find that this lattice consists of the elements T, LZ, SL, B and the elements of the chain Cr which can be defined recursively as follows (see Figure 3): (i) LNB,LRB,LZoRNB,LZoRRB are in Cr, (ii) if V G Cr, then LZo (RZo V) e Cr, (iii) Cr is the smallest sublattice of £(B) satisfying (i) and (ii)....
[...]
...From the description of the lattice of band varieties in [1], [3] or [4] and from [17] it follows that the above-mentioned completely regular semigroup varieties generate the finite lattice of Figure 1....
[...]

Journal Article•DOI•

An Invitation to C-semigroups

[...]

Marcel Jackson¹, Tim E. Stokes²•Institutions (2)

La Trobe University¹, Murdoch University²

01 Jan 2001-Semigroup Forum

TL;DR: In this article, closure semigroups with an additional unary operation called a (right) closure are investigated, and several powerful structural aspects of inverse semigroup theory are shown to extend naturally to some important classes of closure semiigroups.

...read moreread less

Abstract: Semigroups with an additional unary operation called a (right) closure are investigated. These ``closure semigroups'' may be viewed as (not necessarily regular) generalisations of inverse semigroups, and several powerful structural aspects of inverse semigroup theory are shown to extend naturally to some important classes of closure semigroups. These include representations as partial transformations on sets, natural partial orders that are multiplication (and closure) respecting, and simple descriptions of some important congruences.

...read moreread less

68 citations

Cites background from "Varieties of idempotent semigroups"

...varieties has been completely described by several authors ([3], [6], and [10]) and this variety of bands has exactly four proper subvarieties: the trivial variety, the variety SL of semilattices, the variety LZ of left zero semigroups and the variety LN of left normal bands (defined within the variety of bands by the identity xyz = xzy or equivalently by taking the join LZ ∨ SL)....
[...]

Journal Article•DOI•

Finite Semigroups Whose Varieties Have Uncountably Many Subvarieties

[...]

Marcel Jackson¹•Institutions (1)

University of Tasmania¹

15 Jun 2000-Journal of Algebra

TL;DR: For several large classes of semigroups, the authors provides a description of all semigroup which generate varieties with uncountably many sub-varieties, including all Rees quotients of free monoids, the class of finite orthodox monoids.

...read moreread less

53 citations

Cites background from "Varieties of idempotent semigroups"

...The lattice of band varieties has been completely described in [ 1 ], [6] and [7], and it follows that this semigroup generates a variety with only three proper, nontrivial subvarieties (the variety of semilattices, the variety of left zero semigroups and the variety of left normal bands)....
[...]

Collapse

References

PDF

Open Access

More filters

Journal Article•DOI•

Note on idempotent semigroups, IV, Identities of three variables

[...]

Naoki Kimura

01 Jan 1958

10 citations

Varieties of idempotent semigroups

Citations

Cites background from "Varieties of idempotent semigroups"

Cites background from "Varieties of idempotent semigroups"

Cites background from "Varieties of idempotent semigroups"

Cites background from "Varieties of idempotent semigroups"

References

Related Papers (5)