Varieties of operator semigroups representing partitions (Q1899940)

A commutative operator semigroup is an algebra with a unary operation and a binary commutative and associative operation. A subpartition of a set is a partition of a subset of the set. The subpartitions on an \(n\)- element set are represented as \(n\)-ary terms in the paper. The authors give necessary and sufficient conditions on the variety ensuring that there are no constant terms and, for each \(n > 0\), that these representing terms be all the essentially \(n\)-ary terms and moreover that distinct subpartitions yield distinct terms. It is shown that there is precisely one such variety of commutative operator semigroup.