Congruence relations on the lattice of existence varieties of regular semigroups
A class \(\mathcal V\) of regular semigroups is an existence variety (or e-variety) if it is closed under the formation of direct products, regular subsemigroups and homomorphic images. The collection of all e-varieties forms a complete lattice \({\mathcal L}_{ev}\) under inclusion which lattice is the topic of the paper under review. The general line of this investigation is via complete congruences on \({\mathcal L}_{ev}\) which decompose the lattice into disjoint bounded intervals, and idempotent operators on the lattice associated with the congruences. The method of how such complete congruences are found -- radical congruence systems on the underlying class of semigroups -- is merely of universal algebraic nature and is roughly described as follows. Let \(\mathcal C\) be a class of universal algebras; a congruence system \(\rho = \{\rho_S \mid S \in {\mathcal C}\}\) assigns to each member \(S\) of \(\mathcal C\) a congruence \(\rho_S\) on \(S\) in a ``natural way''. (For example, if \(\mathcal C\) consists of all regular semigroups or of all finite semigroups then a typical congruence system is \(\mu\) which assigns to each such semigroup \(S\) the greatest idempotent separating congruence \(\mu_S\) on \(S\).) If \(\mathcal C\) is a variety of universal algebras or the e-variety of all regular semigroups then two types of congruence systems which satisfy some further compatibility conditions are of particular importance (and are called ``(e-)varietal radical congruence system'', respectively ``HS-compatible radical congruence system''). The importance comes from the fact that each such system leads naturally to a complete congruence on the lattice of all sub-(e-)varieties of \(\mathcal C\). (The construction has its origin in some earlier work of Petrich and Reilly concerning the lattice of varieties of completely regular semigroups [\textit{M. Petrich} and \textit{N. R. Reilly}, J. Algebra 134, No. 1, 1-27 (1990; Zbl 0706.20043)].) It is important to note that the present method is ``syntax free'' in the sense that no free object is required for the investigation. For the e-variety \({\mathcal R} {\mathcal S}\) of all regular semigroups, seven systems are presented all of which are HS-compatible radical congruence systems. The central one is the already mentioned system \(\mu\) where \(\mu_S\) is the greatest idempotent separating congruence on the regular semigroup \(S\). Alternatively, \(\mu_S\) may be characterized as the largest congruence on \(S\) all of whose idempotent classes are groups. If we replace ``groups'' by, respectively, ``completely simple semigroups'', ``left groups'', ``right groups'', ``rectangular bands'', ``left zero semigroups'', ``right zero semigroups'' then we get the other six congruence systems. The complex congruences on \({\mathcal L}_{ev}\) associated with these systems are denoted by \(T\) (associated with \(\mu\)) and \(M\), \(T_l\), \(T_r\), \(K\), \(K_l\), \(K_r\), respectively. Many interesting results about these congruences and the associated operators \({\mathcal V} \mapsto {\mathcal V}^T\), etc., are obtained, for instance the equation \({\mathcal E}S = {\mathcal T}^M = \bigvee^\infty_{n = 1} {\mathcal T}^{(T_l T_r)^n}\) where \({\mathcal E} S = E\)-solid semigroups and \({\mathcal I} = \) inverse semigroups. Remarks: At least in the context of the present paper, the distinction between ``HS-compatible radical congruences systems'' and ``e-varietal radical congruence systems'' seems to be superfluous and unnecessarily complicated as all congruence systems under study belong to the former type.
scientific article; zbMATH DE number 840727