Saturation and equational hulls of ordered semigroup varieties
Description
We prove that an inequality-defined variety of partially ordered semigroups is saturated if and only if the ordinary semigroup variety generated by its reducts is saturated. Here saturation excludes proper epimorphic extensions in the ambient category of all semigroups or all partially ordered semigroups. This transfers ordinary equational saturation criteria to compatible partial orders without finiteness, permutation, or convexity assumptions. It recovers a positive answer to Ahanger, Shah, and Khan’s question about ordered normal bands and gives a general equational framework for relaxing convexity in related saturation results. The proof uses the completely regular core E(S)SE(S), a bounded-exponent variety symmetry theorem, and a countable construction for arbitrary inequational theories.