Minimal clans: A class of ordered partial semigroups including boolean rings and lattice-ordered groups

Abstract

The present paper contains a rather comprehensive investigation of the properties of minimal clans — a new class of ordered partial semigroups which includes Boolean rings and lattice-ordered groups as special cases. It is shown that minimal clans preserve many properties of Boolean rings and lattice-ordered groups, that Boolean rings and lattice-ordered groups can be identified as minimal clans having, respectively, a minimal domain of addition or a maximal set of invertible elements, and that minimal clans in turn can be characterized in the classes of all symmetric clans, semiclans, and normal clans. Minimal clans are also compared with some other ordered algebraic structures with partial or complete addition that have been studied in the literature. An example of a minimal clan which is neither a Boolean ring nor a lattice-ordered group is provided by the collection of all fuzzy subsets of a given set.