Partial Kernel Normal Systems for Eventually Regular Semigroups

Shao, Fang; He, Yuan

doi:10.1007/s00233-005-0528-7

Cited by 2 publications

(3 citation statements)

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“…Notice that in [9] the authors showed that any regular congruence ρ on an eventually regular semigroup is uniquely determined by the set of ρ-classes containing idempotents. Using different methods, we generalise this result for regular congruences on a regular-surjective semigroup (see Theorem 2.2, below).…”

Section: The Main Resultsmentioning

confidence: 99%

“…(Note that from [3] it follows that any eventually regular semigroup possesses this property.) Further, it is known that any regular congruence (for which the quotient semigroup induced by this congruence is regular) on an eventually regular semigroup is uniquely determined by: (i) the set of equivalence classes containing idempotents; (ii) its kernel and trace (for the definitions, see after Theorem 2.2) [7,9]. The main aim of this paper is to show that each regular congruence on a regular-surjective semigroup is uniquely determined by the set of equivalence classes containing idempotents, and each regular congruence on an idempotent-regular-surjective semigroup is uniquely determined by its kernel and trace.…”

Section: Introductionmentioning

confidence: 99%

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Regular Congruences on an Idempotent-Regular-Surjective Semigroup

Gigoń¹

2013

Bull. Aust. Math. Soc.

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A semigroup S is called idempotent-surjective (respectively, regular-surjective) if whenever ρ is a congruence on S and aρ is idempotent (respectively, regular) in S /ρ, then there is e ∈ E S ∩ aρ (respectively, r ∈ Reg(S ) ∩ aρ), where E S (respectively, Reg(S )) denotes the set of all idempotents (respectively, regular elements) of S . Moreover, a semigroup S is said to be idempotent-regular-surjective if it is both idempotent-surjective and regular-surjective. We show that any regular congruence on an idempotent-regular-surjective (respectively, regular-surjective) semigroup is uniquely determined by its kernel and trace (respectively, the set of equivalence classes containing idempotents). Finally, we prove that all structurally regular semigroups are idempotent-regular-surjective.2010 Mathematics subject classification: primary 20M99; secondary 06B10.

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Section: The Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Regular Congruences on an Idempotent-Regular-Surjective Semigroup

Gigoń¹

2013

Bull. Aust. Math. Soc.

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“…T α ] is a semilattice of the monoids T α and E = {1 Tα : α ∈ Y } is a subsemigroup of T , then, by Petrich [22], Exercise IV.2 (iv), T is a strong semilattice of T α (α ∈ Y ) with respect to the homomorphism transitive system defined by, for any α, β ∈ E with β α,…”

Section: Preliminariesmentioning

confidence: 98%