Foulis m-semilattices and their modules
Michal Botur, Jan Paseka, Milan Lekár
TL;DR
This work addresses incorporating fuzzy-order semantics into orthomodular lattice theory by organizing endomorphisms as a quantale-like structure and establishing a dagger category framework. It constructs the dagger category $\mathbf{OMLatLin}$ of orthomodular lattices and linear maps, proves that for each lattice $X$ the endomorphisms $\mathbf{Lin}(X)$ form a Foulis m-semilattice (a quantale-like object), and shows that $X$ is a left $\mathbf{Lin}(X)$-module. It further develops a dual viewpoint via the Sasaki-projection lattice $[S]$ of a Foulis m-semilattice $S$, which also carries a left $S$-module and exhibits a Sasaki action $\sigma_u$ that is self-adjoint, linear, and idempotent with image $\downarrow u$. Collectively, these results fuse orthomodular lattice theory with quantale/module machinery and introduce a fuzzy-theoretic lens for quantum-logical structures, with proposed future work on dagger kernels and semi-quantaloid structures.
Abstract
Building upon the results of Jacobs, we show that the category OMLatLin of orthomodular lattices and linear maps forms a dagger category. For each orthomodular lattice X, we construct a Foulis m-semilattice Lin(X) composed of endomorphisms of X. This m-semilattice acts as a quantale, enabling us to regard X as a left Lin(X)-module. Our novel approach introduces a fuzzy-theoretic dimension to the theory of orthomodular lattices.