The theory of residuated lattices, first proposed by Ward and Dilworth [4], is formalised in Isabelle/HOL. This includes concepts of residuated functions; their adjoints and conjugates. It also contains necessary and sufficient conditions for the existence of these operations in an arbitrary lattice. The mathematical components for residuated lattices are linked to the AFP entry for relation al...

Let us say that a class of upward closed sets (upsets) distributive lattices is finitary filter if it under homomorphic preimages, intersections, and directed unions. We show the only classes upsets are formed by what we call n-filters. These related to finite Boolean lattice with n atoms in same way filters two-element lattice: n-filters precisely intersections prime pre-images n-filter non-ze...

This paper presents the algebraic and Kripke model soundness and completeness of a logic over Boolean monoids. An additional axiom added to the logic will cause the resulting monoid models to be representable as monoids of relations. A star operator, interpreted as reflexive, transitive closure, is conservatively added to the logic. The star operator is a relative modal operator, i.e., one that...

We give a complete classification of the factorial functions of Eulerian binomial posets. The factorial function B(n) either coincides with n!, the factorial function of the infinite Boolean algebra, or 2n−1, the factorial function of the infinite butterfly poset. We also classify the factorial functions for Eulerian Sheffer posets. An Eulerian Sheffer poset with binomial factorial function B(n...

We completely characterize the factorial functions of Eulerian binomial posets. The factorial function B(n) either coincides with n!, the factorial function of the infinite Boolean algebra, or 2n−1, the factorial function of the infinite butterfly poset. We also classify the factorial functions for Eulerian Sheffer posets. An Eulerian Sheffer poset with binomial factorial function B(n) = n! has...

We introduce a certain class of algebras associated to matroids. We prove the Lefschetz property of the algebras for some special cases. Our result implies the Sperner property for the Boolean lattice and the vector space lattice. Résumé. Nous présentons une classe d’algèbres associées aux matroı̈des. Nous démontrons que dans quelques cas spécifiques, ces algèbres verifient la propriété de Lefsc...