Irreducible varieties of commutative semigroups
This paper is based on a complete structural description of the lattice of varieties of commutative semigroups which is due to \textit{A. Kisielewicz} [Trans. Am. Math. Soc. 342, No. 1, 275-306 (1994; Zbl 0801.20042)]: each variety corresponds uniquely to a certain quadruple \((J,m,r,\pi)\), where \(\pi\) is a so-called `remainder' of the triple \((J,m,r)\). In a previous joint paper with \textit{A. Kisielewicz} [J. Algebra 232, No. 2, 493-506 (2000; Zbl 0970.20033)], the author has used this structural description to effectively characterize the covers and the dual covers of a given variety in the lattice. In the present paper, through a finer combinatorial analysis which leads to a proof of the existence of unique minimal generators for remainders, those characterizations are further refined by describing covers and dual covers in lattices of remainders. This allows the author to give a complete description of both join- and meet-irreducibles in the lattice of varieties of commutative semigroups. As a simple application, he deduces that the variety of all commutative semigroups is the only element of this lattice which is both join- and meet-irreducible.
scientific article; zbMATH DE number 1901961