On the Structure of Balanced Residuated Partially Ordered Monoids

A residuated poset is a structure \(\langle A,\leqslant ,\cdot ,\backslash ,/,1\rangle \) where \(\langle A,\leqslant \rangle \) is a poset and \(\langle A,\cdot ,1\rangle \) is a monoid such that the residuation law \(x\cdot y\leqslant z\iff x\leqslant z/y\iff y\leqslant x\backslash z\) holds. A residuated poset is balanced if it satisfies the identity \(x\backslash x \approx x/x\) . 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.