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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:
  PETRICH M.: Introduction to Semigroups. Merrill, Columbus, 1973. · Zbl 0321.20037  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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