Generalized continuous and left-continuous t-norms arising from algebraic semantics for fuzzy logics

This paper focuses on the issue of how generalizations of continuous and left-continuous t-norms over linearly ordered sets should be from a logical point of view. Taking into account recent results in the scope of algebraic semantics for fuzzy logics over chains with a monoidal residuated operation, we advocate linearly ordered BL-algebras and MTL-algebras as adequate generalizations of continuous and left-continuous t-norms respectively. In both cases, the underlying basic structure is that of linearly ordered residuated lattices. Although the residuation property is equivalent to left-continuity in t-norms, continuous t-norms have received much more attention due to their simpler structure. We review their complete description in terms of ordinal sums and discuss the problem of describing the structure of their generalization to BL-chains. In particular we show the good behavior of BL-algebras over a finite or complete chain, and discuss the partial knowledge of rational BL-chains. Then we move to the general non-continuous case corresponding to left-continuous t-norms and MTL-chains. The unsolved problem of describing the structure of left-continuous t-norms is presented together with a fistful of construction-decomposition techniques that apply to some distinguished families of t-norms and, finally, we discuss the situation in the general study of MTL-chains as a natural generalization of left-continuous t-norms

Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality

We study representations of MV-algebras -- equivalently, unital lattice-ordered abelian groups -- through the lens of Stone-Priestley duality, using canonical extensions as an essential tool. Specifically, the theory of canonical extensions implies that the (Stone-Priestley) dual spaces of MV-algebras carry the structure of topological partial commutative ordered semigroups. We use this structure to obtain two different decompositions of such spaces, one indexed over the prime MV-spectrum, the other over the maximal MV-spectrum. These decompositions yield sheaf representations of MV-algebras, using a new and purely duality-theoretic result that relates certain sheaf representations of distributive lattices to decompositions of their dual spaces. Importantly, the proofs of the MV-algebraic representation theorems that we obtain in this way are distinguished from the existing work on this topic by the following features: (1) we use only basic algebraic facts about MV-algebras; (2) we show that the two aforementioned sheaf representations are special cases of a common result, with potential for generalizations; and (3) we show that these results are strongly related to the structure of the Stone-Priestley duals of MV-algebras. In addition, using our analysis of these decompositions, we prove that MV-algebras with isomorphic underlying lattices have homeomorphic maximal MV-spectra. This result is an MV-algebraic generalization of a classical theorem by Kaplansky stating that two compact Hausdorff spaces are homeomorphic if, and only if, the lattices of continuous [0, 1]-valued functions on the spaces are isomorphic.Comment: 36 pages, 1 tabl

The pre-Lie structure of the time-ordered exponential

The usual time-ordering operation and the corresponding time-ordered exponential play a fundamental role in physics and applied mathematics. In this work we study a new approach to the understanding of time-ordering relying on recent progress made in the context of enveloping algebras of pre-Lie algebras. Various general formulas for pre-Lie and Rota-Baxter algebras are obtained in the process. Among others, we recover the noncommutative analog of the classical Bohnenblust-Spitzer formula, and get explicit formulae for operator products of time-ordered exponentials

A non-commutative Priestley duality

We prove that the category of left-handed strongly distributive skew lattices with zero and proper homomorphisms is dually equivalent to a category of sheaves over local Priestley spaces. Our result thus provides a non-commutative version of classical Priestley duality for distributive lattices and generalizes the recent development of Stone duality for skew Boolean algebras. From the point of view of skew lattices, Leech showed early on that any strongly distributive skew lattice can be embedded in the skew lattice of partial functions on some set with the operations being given by restriction and so-called override. Our duality shows that there is a canonical choice for this embedding. Conversely, from the point of view of sheaves over Boolean spaces, our results show that skew lattices correspond to Priestley orders on these spaces and that skew lattice structures are naturally appropriate in any setting involving sheaves over Priestley spaces.Comment: 20 page