Let $S=\{x_1,\cdots ,x_n\}$ be a finite subset of a partially ordered set $P$. Let $f$ be an incidence function of $P$. Let $\left[f(x_i\wedge x_j)\right]$ denote the $n\times n$ matrix having $f$ evaluated at the meet $x_i\wedge x_j$ of $x_i$ and $x_j$ as its $i,j$-entry and $\left[f(x_i\vee x_j)\right]$ denote the $n\times n$ matrix having $f$ evaluated at the join $x_i\vee x_j$ of $x_i$ and $x_j$ as its $i,j$-entry. The set $S$ is said to be meet-closed if $x_i\wedge x_j\in S$ for all $1\le i,j\le n$. In this paper we get explicit combinatorial formulas for the determinants of matrices $\left[f(x_i\wedge x_j)\right]$ and $\left[f(x_i\vee x_j)\right]$ on any meet-closed set $S$. We also obtain necessary and sufficient conditions for the matrices...

We investigate $\sqcap$-directoids which are bounded and equipped by a unary operation which is an antitone involution. Hence, a new operation $\bigsqcup$ can be introduced via De Morgan laws. Basic properties of these algebras are established. On every such an algebra a ring-like structure can be derived whose axioms are similar to that of a generalized boolean quasiring. We introduce a concept of symmetrical difference and prove its basic properties. Finally, we study conditions of direct decomposability of directoids...

It is well-known that every MV-algebra is a distributive lattice with respect to the induced order. Replacing this lattice by the so-called directoid (introduced by J. Ježek and R. Quackenbush) we obtain a weaker structure, the so-called skew MV-algebra. The paper is devoted to the axiomatization of skew MV-algebras, their properties and a description of the induced implication algebras.

It is shown that every directoid equipped with sectionally switching mappings can be represented as a certain implication algebra. Moreover, if the directoid is also commutative, the corresponding implication algebra is defined by four simple identities.

We modify slightly the definition of $H$-partial functions given by Celani and Montangie (2012); these partial functions are the morphisms in the category of $H^{\vee }$-space and this category is the dual category of the category with objects the Hilbert algebras with supremum and morphisms, the algebraic homomorphisms. As an application we show that finite pure Hilbert algebras with supremum are determined by the monoid of their endomorphisms.