On the structure of balanced residuated partially ordered monoids

A residuated poset is a structure $⟨A,\le ,\cdot ,\mathrm{\backslash },\mathrm{/},1⟩$ where $⟨A,\le ⟩$ is a poset and $⟨A,\cdot ,1⟩$ is a monoid such that the residuation law $x\cdot y\le z⟺x\le z\mathrm{/}y⟺y\le x\mathrm{\backslash }z$ holds. A residuated poset is balanced if it satisfies the identity $x\mathrm{\backslash }x\approx x\mathrm{/}x$. By generalizing the well-known construction of Plonka sums, we show that a specific class of balanced residuated posets can be decomposed into such a sum indexed by the set of positive idempotent elements. Conversely, given a semilattice directed system of residuated posets equipped with two families of maps (instead of one, as in the usual case), we construct a residuated poset based on the disjoint union of their domains. We apply this approach to provide a structural description of some varieties of residuated lattices and relation algebras.