Irreducible varieties of commutative semigroups

Automorphisms of the lattice of equational theories of commutative semigroups

“…We define B ≤ C if and only if f (B) ⊆ C for some f generated by g ij , h 1 and permutations. We note that under this quasi-ordering, B and C are equivalent if and only if p(B) = C for some permutation p (see [1]). …”

“…Observe that g 1,3 (b), g 1,2 (b) ∈ J when s ≤ 2 and s = 1, respectively. Similarly for the second case.…”

“…In the first case, c ≥ a and a ≤ c, and E(a, c) is definable by Lemma 6.5 (2b − 2 is the largest term, occurring exactly once, in this case). Moreover, we know that a 1 …”

Definability in the lattice of equational theories of commutative semigroups

“…These sets play independent roles in the proofs. However, in the final presentation, (N1, N2) can be omitted, since, as it was observed by M. Grech in [1], (N1, N2) follow from (π3, π4). That is why we have decided to replace here (N1-N4) by (C r ) and (C m ).…”

“…Algorithms and formulas in [11] provide a general method of computing in equational theories and varieties of commutative semigroups. This method has already proved to be useful in solving a number of problems [3,1,2,12,13]. In this paper we use it to solve the problem suggested by McKenzie.…”