Certain congruences on orthodox semigroups

Mills, Janet E.

doi:10.2140/pjm.1976.64.217

Cited by 7 publications

(5 citation statements)

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“…The inverse semigroup S is said to be strongly E-reflexive if, given e G £ ' and x and y in S, exy G E implies that eyx G E, where the element 1 is the identity of S 1 . Note that the subdirect product of a family of strongly E-reflexive inverse semigroups is again strongly E-reflexive.…”

Section: Proposition 3 (See 15) Let a Be An Tj-class Of S Then The mentioning

confidence: 99%

“…The inverse semigroup S is said to be strongly E-reflexive if, given e G £ ' and x and y in S, exy G E implies that eyx G E, where the element 1 is the identity of S 1 .…”

Section: Proposition 3 (See 15)mentioning

confidence: 99%

“…Suppose that L m is the semilattice A of £-unitary inverse semigroups 5 A , A G A, where (b, g) G S M say, g being the non-identity element of Gp. Then (c,«) = ( i , g r ' £ S , so that (c, e p ) = (c, g){c, g)' 1 This example is a variant on (13, Example 5.2).…”

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confidence: 96%

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Strongly E-reflexive inverse semigroups II

O’Carroll

1978

Proceedings of the Edinburgh Mathematical Society

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In a recent paper (13), we introduced the class of strongly jE-reflexive inverse semigroups. This class was shown to coincide with the class of those inverse semigroups which are semilattices of E-unitary inverse semigroups. In particular, therefore, E-unitary inverse semigroups and semilattices of groups are stronglŷ -reflexive, and in fact so are subdirect products of these two types of semigroups. J. Mills (1) considers orthodox semigroups which are subdirect products of an £-unitary regular semigroup and a semilattice of groups, and of course there are strong connections between the two papers.In this communication we wish in part to .specialise Mills' results to inverse semigroups, and in doing so, to consider them in the context of the theory of strongly E-reflexive inverse semigroups. As well, we give yet another characterisation of these semigroups in terms of JS-unitary inverse semigroups, and show that an inverse semigroup which is a semilattice of strongly jE-reflexive inverse semigroups is again strongly E-reflexive.In this section, some results on congruences which will be needed below are collected together.Let S be an inverse semigroup with semilattice of idempotents E. Let a = {(*, y) £ S x S\ex -ey for some e G E}; then a is the minimum group congruence on S (10). Moreover, S is said to be E-unitary if Ea = E. In general, there exists a minimum .E-unitary congruence on S, which we shall denote by K. Proposition 1. (See 12) #c is the congruence on S generated by a D£%. Let n = {(a, b)G S x S\a'] ea = b~leb for all e £ £ ) ; then ft is the maximum idempotent-separating congruence on S (7). Recall that a congruence v on S is called idempotent-determined (5) if (e, x)Gv and eE.E imply that xGE; as is natural, a homomorphism on S will be called idempotent-determined when the associated congruence on S is so. The minimum semilattice of groups congruence on S will be denoted by £, the minimum semilattice congruence by 17, and the identity congruence by 1.Note that K C 17 C\a and that £ C 17 Ha.

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Section: Proposition 3 (See 15) Let a Be An Tj-class Of S Then The mentioning

confidence: 99%

“…The inverse semigroup S is said to be strongly E-reflexive if, given e G £ ' and x and y in S, exy G E implies that eyx G E, where the element 1 is the identity of S 1 .…”

Section: Proposition 3 (See 15)mentioning

confidence: 99%

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Strongly E-reflexive inverse semigroups II

O’Carroll

1978

Proceedings of the Edinburgh Mathematical Society

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“…A regular semigroup S such that E(S) is a subsemigroup is called orthodox. Hall [3] - [5], Meakin [10,11], Yamada [13] and many others like Milles [12], McAlister [9] have studied orthodox semigroups and characterized such semigroup S by E(S). Another approach, introduced by C. Eberhart, W. Williams and L. Kinch [2] is to study a semigroup S by the subsemigroup generated by E(S).…”

Section: Introductionmentioning

confidence: 99%

On the subsemigroup generated by ordered idempotents of a regular semigroup

Bhuniya

Hansda

2015

Discuss. Math. - General Algebra and Applications

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An element e of an ordered semigroup S is called an ordered idempotent if e ≤ e 2. Here we characterize the subsemigroup < E ≤ (S) > generated by the set of all ordered idempotents of a regular ordered semigroup S. If S is a regular ordered semigroup then < E ≤ (S) > is also regular. If S is a regular ordered semigroup generated by its ordered idempotents then every ideal of S is generated as a subsemigroup by ordered idempotents.

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“…Pastijn and Petrich [11] characterized the least and the greatest congruences on a regular semigroup S with a given trace and kernel. For an inverse semigroup, and more generally for an orthodox semigroup S, the least Clifford congruence was described by Mills [10]. LaTorre [8] used the least group congruence on a regular semigroup given by Feigenbaum [2] to describe the least Clifford congruence ξ on a regular semigroup.…”

Section: Introductionmentioning

confidence: 99%