Lattices of completely regular semigroup varieties

Pastijn, Francis; Trotter, Peter

doi:10.2140/pjm.1985.119.191

Cited by 46 publications

(35 citation statements)

References 19 publications

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“…The variety V* was denoted by V. . mm in [13]. We now point out the connection between the above results and the results of [16].…”

Section: The Relation K Which Is Given Bysupporting

confidence: 57%

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The lattice of completely regular semigroup varieties

Pastijn

1990

J Aust Math Soc A

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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 £(CR) of completely regular semigroup varieties we define two closure operations which induce complete congruences. The consideration of a third complete congruence on £(CR) yields a subdirect decomposition of £(CR). Using these results we show that £(CR) is arguesian. This confirms the (tacit) conjecture that £(CR) is modular.

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“…The variety V* was denoted by V. . mm in [13]. We now point out the connection between the above results and the results of [16].…”

Section: The Relation K Which Is Given Bysupporting

confidence: 57%

“…From this it also follows that the relation T on £(CR) can be denned by V r \ V « G o V = GoW. The variety V r was denoted by V (/v) in [13] and by V + in [19]. Using …”

Section: The Relation T = T { C\t R Is Again a Complete Congruence Anmentioning

confidence: 98%

The lattice of completely regular semigroup varieties

Pastijn

1990

J Aust Math Soc A

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“…COROLLARY [9,THEOREM 3.2]. If p and 9 are congruences on S, then tr pCtxd ^ ptÇOt, pTçeT, ker p Ç ker 0 => pk Q Ok ■…”

Section: 13mentioning

confidence: 99%

Congruences on Regular Semigroups

Pastijn¹,

Petrich²

1986

Transactions of the American Mathematical Society

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ABSTRACT. Let S be a regular semigroup and let p be a congruence relation on S. The kernel of p, in notation kerp, is the union of the idempotent p-classes. The trace of p, in notation trp, is the restriction of p to the set of idempotents of S. The pair (kerp,trp) is said to be the congruence pair associated with p. Congruence pairs can be characterized abstractly, and it turns out that a congruence is uniquely determined by its associated congruence pair. The triple ((pV £■)/£, kerp, (pV %)/%) is said to be the congruence triple associated with p. Congruence triples can be characterized abstractly and again a congruence relation is uniquely determined by its associated triple.The consideration of the parameters which appear in the above-mentioned representations of congruence relations gives insight into the structure of the congruence lattice of S. Introduction and summary. After it was realized by Wagner that a congruence on an inverse semigroup S is uniquely determined by its idempotent classes, Preston provided an abstract characterization of such a family of subsets of S called the kernel normal system (see [2, Chapter 10]). This approach was the only usable means for handling congruences on inverse semigroups for two decades. A new approach to the problem of describing congruences on inverse semigroups was sparked by the work of Scheiblich [13] who described congruences in terms of kernels and traces. A systematic exposition of the achievements of this approach can be found

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“…This section is devoted to a remarkable example, due to P. G. Trotter, of a relatively free inverse semigroup which is not completely semisimple. [5], the congurence p generated by 6 is a fully invariant congruence. Now py is the smallest fully invariant congruence on Fix containing (aa"1 ,o?a~2) so that p-y C p. On the other hand, 6 C p-y so that p C p-y.…”

Section: 1 (Aaa~l) £ P(f') Hence Kerp(f') = E(f') Since Tr/z(f'mentioning

confidence: 99%

Properties of relatively free inverse semigroups

Reilly¹,

Trotter²

1986

Trans. Amer. Math. Soc.

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ABSTRACT.The objective of this paper is to study structural properties of relatively free inverse semigroups in varieties of inverse semigroups. It is shown, for example, that if S is combinatorial (i.e., X is trivial), completely semisimple (i.e., every principal factor is a Brandt semigroup or, equivalently, S does not contain a copy of the bicyclic semigroup) or F-unitary (i.e., E(S) is the kernel of the minimum group congruence) then the relatively free inverse semigroup F"Vx on the set X in the variety "V generated by S is also combinatorial, completely semisimple or F-unitary, respectively.If 5 is a fundamental (i.e., the only congruence contained in M is the identitycongruence) and \X\ > No, then FVx ia a'so fundamental.FVx may not be fundamental if |^f| < No-It is also shown that for any variety of groups U and for |X| > No, there exists a variety of inverse semigroups "V which is minimal with respect to the properties (i) FVx ls fundamental and (ii) "V n Q = U, where Q is the variety of groups.In the main result of the paper it is shown that there exists a variety V for which FVx 's not completely semisimple, thereby refuting a long standing conjecture. Summary.In general, the relatively free objects in any variety of algebras are important in the study of that variety and this has been true, in particular, in the study of inverse semigroups. The first good description of the free inverse semigroup on one generator was given by Gluskin [2], while the first good description of the free inverse semigroup on an arbitrary set was given by Scheiblich [12]. For a survey of these and related results, see Petrich [6] and Reilly [10].The structures of the free group, the free inverse semigroup and free semilattice of groups are well known. However, with the one exception of the work on the relatively free objects in varieties generated by £"-unitary inverse semigroups by Pastijn [4] and Petrich and Reilly [7], relatively little has been said regarding the structure of the relatively free objects in other varieties of inverse semigroups or with regard to other properties than ^-unitary.Since it would seem to be beyond the present state of knowledge to give explicit structure theorems for nontrivial relatively free inverse semigroups other than those considered by Gluskin and Scheiblich, we continue the investigation of the structure of relatively free inverse semigroups in the spirit of Pastijn [4] and Petrich and Reilly [7]- §2 is devoted to background information. In §3, it is shown that certain structural properties of an inverse semigroup S will be inherited by the relatively free objects F"Vx, in the variety ~V generated by X. If 5 is combinatorial or completely

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Lattices of completely regular semigroup varieties

Cited by 46 publications

References 19 publications

The lattice of completely regular semigroup varieties

The lattice of completely regular semigroup varieties

Congruences on Regular Semigroups

Properties of relatively free inverse semigroups

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