On the structure of balanced residuated partially ordered monoids

A residuated poset is a structure ⟨A,⩽,·,\,/,1⟩ where ⟨A,⩽⟩ is a poset and ⟨A,·,1⟩ is a monoid such that the residuation law x·y⩽z⟺x⩽z/y⟺y⩽x\z holds. A residuated poset is balanced if it satisfies the identity x\x≈x/x. By generalizing the well-known construction of Płonka 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.