We develop a theory of sets with distributive products (called shelves and multi-shelves) and of their homology. We relate the shelf homology to the rack and quandle homology. 1. Binary distributive operations Let X be a set. Binary operations ? : X ×X → X with composition x ?1 ?2y = (x ?1 y) ?2 y and the identity x ? y = x form a monoid, M(X). A collection of operations ?i, i ∈ I is called mut...

A NOTE ON SUBMODULAR FUNCTIONS ON DISTRIBUTIVE LATTICES Satoru Fujishige University of Tsukuba Nobuaki Tomizawa Niigata University (Received July 15, 1982; Rllvised June 14, 1983) Let D be a distributive lattice formed by subsets of a finite set E with 1/>, E E D and let R be the set of reals. Also let f be a submodular function from D into R with f(l/» = O. We determine the set of extreme poin...

Based on a complete Heyting algebra, we modify the definition oflattice-valued fuzzifying convergence space using fuzzy inclusionorder and construct in this way a Cartesian-closed category, calledthe category of $L-$ordered fuzzifying convergence spaces, in whichthe category of $L-$fuzzifying topological spaces can be embedded.In addition, two new categories are introduced, which are called the...

A computable presentation of the linearly ordered set (ω,≤), where ω is the set of natural numbers and ≤ is the natural order on ω, is any linearly ordered set L = (ω,≤L) isomorphic to (ω,≤) such that ≤L is a computable relation. Let X be subset of ω and XL be the image of X in the linear order L under the isomorphism between (ω,≤) and L. The degree spectrum of X is the set of all Turing degree...

In this paper, we introduce and study the stable topology on the set of prime filters of a bounded 0-distributive lattice. The stable topology is a subtopology of the hull kernel topology on the set of prime filters of a bounded 0-distributive lattice. Sufficient condition is given under which the hull kernel topology and stable topology coincide on the set of prime filters (the set of maximal ...

A very simple proof of the finite embeddability property for residuated distributivelattice-ordered groupoids and some related classes of algebras is presented. In particular, this gives an answer to the question posed in [3, Problem 4.2]. The presented construction allows for improvement of the upper bound on the complexity of the decision procedure for the universal theory of residuated distr...

We study Nonassociative Lambek Calculus with additives ∧,∨, satisfying the distributive law (Distributive Full Nonassociative Lambek Calculus DFNL). We prove that categorial grammars based on DFNL, also enriched with assumptions, generate context-free languages. The proof uses proof-theoretic tools (interpolation) and a construction of a finite model, earlier employed in [11] in the proof of Fi...