Abstract
All varieties of idempotent semigroups have been classified with respect to the unification types of their defining sets of identities. With the exception of eight finitary unifying theories, they are all of unification type zero. This yields countably many examples of theories of this type which are more “natural” than the first example constructed by Fages and Huet.
The lattice of all varieties of idempotent semigroups is a sublattice of the lattice of all varieties of orthodox bands of groups, and this lattice is a sublattice of the lattice of all varieties of completely regular semigroups. The proof which was used to establish the result for the varieties of idempotent semigroups of type zero can—with some modifications—also be applied to the larger lattice of all varieties of completely regular semigroups. This shows that type zero is not an exception, but rather common for varieties of semigroups.
To establish the results for the eight exceptional finitary varieties of idempotent semigroups we have developed a method which under certain conditions allows to deduce the unification type of a join of varieties from the types of the varieties participating in this join. This method can also be employed for varieties of orthodox bands of abelian groups. Any variety of orthodox bands of abelian groups is the join of a variety of idempotent semigroups and a variety of abelian groups. It turns out that the unification type of such a join is just the type of the variety of idempotent semigroups taking part in this join.
The emphasis of the paper is on describing the tools necessary for proving all the mentioned results.
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© 1992 Springer-Verlag Berlin Heidelberg
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Baader, F. (1992). Unification in varieties of completely regular semigroups. In: Schulz, K.U. (eds) Word Equations and Related Topics. IWWERT 1990. Lecture Notes in Computer Science, vol 572. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-55124-7_9
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DOI: https://doi.org/10.1007/3-540-55124-7_9
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