A partial ordering P is said to have the weak Freese-Nation property (WFN) if there is a mapping f : P → [P ]0 such that, for any a, b ∈ P , if a ≤ b then there exists c ∈ f(a) ∩ f(b) such that a ≤ c ≤ b. In this note, we study the WFN and some of its generalizations. Some features of the class of Boolean algebras with the WFN seem to be quite sensitive to additional axioms of set theory: e.g.,...

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A partial ordering P is said to have the weak Freese-Nation property (WFN) if there is a mapping f : P → [P ]0 such that, for any a, b ∈ P , if a ≤ b then there exists c ∈ f(a) ∩ f(b) such that a ≤ c ≤ b. In this note, we study the WFN and some of its generalizations. Some features of the class of Boolean algebras with the WFN seem to be quite sensitive to additional axioms of set theory: e.g.,...

The expressiveness of Boolean Algebras is significantly extended by the Boolean Differential Calculus (BDC). The additionally defined differentials of Boolean variables, differentials and further differential operators of Boolean functions as well as several derivative operations of Boolean functions allow to model changes of function values together with changes of the values of variables and ...

This paper is the second of a two part series. In this Part, we prove, using the description of simples obtained in Part I, that the variety $mathbf{RDQDStSH_1}$ of regular dually quasi-De Morgan Stone semi-Heyting algebras of level 1 is the join of the variety generated by the twenty 3-element $mathbf{RDQDStSH_1}$-chains and the variety of dually quasi-De Morgan Boolean semi-Heyting algebras--...

In Boolean algebra, true statements are denoted 1 and false statements are denoted 0. A Boolean function acts on a set of these Boolean values and outputs a set of Boolean values (usually just one). The most common Boolean operators used are NOT, AND, OR, and XOR. 2.1 NOT NOT is a Boolean function that takes in one Boolean value and outputs its negation. Let x be a Boolean variable. NOT(x) is d...