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The Lattice of Equational Classes of Commutative Semigroups

1971; Cambridge University Press; Volume: 23; Issue: 5 Linguagem: Inglês

10.4153/cjm-1971-098-0

1496-4279

Evelyn Nelson,

Rough Sets and Fuzzy Logic

There has been some interest lately in equational classes of commutative semigroups (see, for example, [ 2; 4; 7; 8 ]). The atoms of the lattice of equational classes of commutative semigroups have been known for some time [ 5 ]. Perkins [ 6 ] has shown that each equational class of commutative semigroups is finitely based. Recently, Schwabauer [ 7; 8 ] proved that the lattice is not modular, and described a distributive sublattice of the lattice. The present paper describes a “skeleton” sublattice of the lattice, which is isomorphic to A × N + with a unit adjoined, where A is the lattice of pairs ( r, s ) of non-negative integers with r ≦ s and s ≧ 1, ordered component-wise, and N + is the natural numbers with division.