Residual coherentization of balanced residuated partially ordered semigroups

Abstract

This paper develops a canonical decomposition--reconstruction theory for balanced residuated partially ordered semigroups. The starting point is the intrinsic local-unit map $τ(x)=x\backslash x=x/x, $ whose values are positive idempotents. The primitive fibres of this map are generally too fine to be compatible with multiplication and residuals: the local units of $xy$, $x\backslash y$, and $x/y$ need not be determined by the local units of $x$ and $y$. We therefore construct the residual coherentization $\mathcal C_{\mathrm r}(\mathbf M)$, the finest quotient of the positive-idempotent skeleton on which these three local-unit outputs are well defined at quotient level. The blocks of $\mathcal C_{\mathrm r}(\mathbf M)$ define the canonical components, while the quotient skeleton records the target component for products and residuals. Together with the component algebras, the product-shadow maps $x\mapsto xq$, and the residual-shadow maps $y\mapsto y/q$, these data reconstruct the original algebra. The final part compares this construction with subsemilattice-steady visibility decompositions. At every finite stage of the induced iterative decompositions, the partition obtained from residual coherentization is finer than the partition obtained from any subsemilattice-steady visibility choice. Equivalently, each component produced by the subsemilattice-steady construction is a union of residual-coherent components.