Some properties and characterizations for abundant semigroups with\n generalised quasiadequate transversals are explored. In such semigroups, an\n interesting property [?a,b ? Re1S, VSo(a) ? VSo (b) ? 0 ? VSo (a) = VSo\n (b)] is investigated and thus the concept of refined generalised\n quasi-adequate transversals, for short, RGQA transversals is introduced. It\n is shown that RGQA transversals are the real common generalisations of both\n orthodox transversals and adequate transversals in the abundant case.\n Finally, by means of two abundant semigroups R and L, a spined product\n structure theorem for an abundant semigroup with a quasi-ideal RGQA\n transversal is established.

/pdf/on-refined-generalised-quasi-adequate-transversals-3r0fza8i27.pdf

On refined generalised quasi-adequate transversals

As the real common generalisations of both orthodox transversals and adequate
 transversals in abundant semigroups, the concept of refined generalised
 quasi-adequate transversals, briefly, RGQA transversals was introduced by
 Kong and Wang. In this paper, for the RGQA transversal, the necessary and
 sufficient condition for the sets I and ? to be bands is investigated. It is
 demonstrated that the sets I and ? are both bands if and only if the RGQA
 transversal is weakly simplistic. Moreover, the RGQA transversal So being
 weakly simplistic is different from So being a quasi-ideal nor the abundant
 semigroup S satisfying the regularity condition. Finally, by means of a
 quasi-adequate semigroup and a band, the structure theorem for an abundant
 semigroup with a weakly simplistic RGQA transversal is established.

/pdf/abundant-semigroups-with-weakly-simplistic-rgqa-transversals-3r01u98o.pdf

Abundant semigroups with weakly simplistic RGQA transversals

/pdf/quasi-ideal-ehresmann-transversals-the-spined-product-5121r32a2n.pdf

Quasi-ideal Ehresmann transversals: The spined product structure

Abstract As the real common generalisations of both orthodox transversals and adequate transversals in the abundant case, the concept of refined generalised quasi-adequate transversal, for short, RGQA transversal was introduced by Kong and Wang. In this paper, an interesting characterization for a generalised quasi-adequate transversal to be refined is acquired. It is shown that the product of every two quasi-ideal RGQA transversals of the abundant semigroup S satisfying the regularity condition is a quasi-ideal RGQA transversal of S and that all quasi-ideal RGQA transversals of S compose a rectangular band. The related results concerning adequate transversals are generalised and enriched.

/pdf/the-product-of-quasi-ideal-refined-generalised-quasi-4h9ph3dn5b.pdf

The product of quasi-ideal refined generalised quasi-adequate transversals

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The product of quasi-ideal refined generalised quasi-adequate transversals

Quasi-ideal Ehresmann transversals: The spined product structure