On -semigroups
In the paper, the following concept are defined: (i) a minimal left (right, two-sided) ideal with respect to a subset of a semigroup , (ii) a kernel with respect to a subset of a semigroup , and their basic properties are investigated.
A semigroup variety is called modular if it is a modular element of the lattice of all semigroup varieties. We obtain a strong necessary condition for a semigroup variety to be modular. In particular, we prove that every modular nil-variety may be given by 0-reduced identities and substitutive identities only. (An identity is called substitutive if the words and depend on the same letters and may be obtained from by renaming of letters.) We completely determine all commutative modular...
2000 Mathematics Subject Classification: 20M20, 20M10.We describe maximal nilpotent subsemigroups of a given nilpotency class in the semigroup Ωn of all n × n real matrices with non-negative coefficients and the semigroup Dn of all doubly stochastic real matrices.