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The translational hull of a normal cryptogroup. (English) Zbl 0807.20051
A normal cryptogroup is a completely regular semigroup \(S\) in which \(\mathcal H\) is a congruence and \(S/{\mathcal H}\) is a normal band. Since such semigroups are precisely those semigroups which can be expressed as strong semilattices of completely simple semigroups, one can set \(S = [Y; S_ \alpha, \chi_{\alpha \beta}]\), where \(Y\) is a semilattice, \(S_ \alpha\) is a completely simple semigroup for each \(\alpha \in Y\) and \(\chi_{\alpha \beta} : S_ \alpha \to S_ \beta\) for \(\alpha \geq \beta\) are structure homomorphisms. The translational hull of \(S\) is constructed in terms of the translational hull of the underlying semilattice \(Y\) and the translational hulls of the completely simple components \(S_ \alpha\). This construction is specialized to regular semigroups which form a subdirect product of a semilattice and a completely simple semigroup.
Further considerations concern the threads in a strong semilattice corresponding to ideals in \(Y\). The set \(\mathcal T\) of all threads is closed under componentwise multiplication on the common subset of indices. The semigroup \(\mathcal T\) is used to characterize the ideals \(\Omega_ i(S)\) of the translational hull \(\Omega(S)\) corresponding to inner bitranslations of the components \(S_ \alpha\) and retract ideals of \(Y\) and to establish some properties of \(\Omega_ i(S)\) including statements concerning its position within \(\Omega(S)\) for a normal cryptogroup \(S\).

MSC:
20M10 General structure theory for semigroups
20M17 Regular semigroups
06A12 Semilattices
20M20 Semigroups of transformations, relations, partitions, etc.
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References:
[1] PETRICH M.: Introduction to Semigroups. Merrill, Columbus, 1973. · Zbl 0321.20037
[2] PETRICH M.: The translational hull of a semilattice of weakly reductive semigroups. Canad. J. Math. 26 (1974), 1520-1536. · Zbl 0303.20049
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