Annals of Communications in Mathematics

Abstract:This paper studies \( B \)-algebras through the Boolean-centre-valued operators \( x^\nabla = 1 \to x \) and \( x^\Delta = 0 \leftarrow x \). The operator \( {}^\Delta \) defines a centre-valued negation satisfying \( (x \vee y)^\Delta = x^\Delta \wedge y^\Delta \) for all \( x,y \in A \), together with the inequality \( x^\Delta \vee y^\Delta \leq (x \wedge y)^\Delta \). Equality in the latter holds on the Boolean centre \( B(A) \); an explicit example confirms that restricting to \( B(A) \) is necessary. The Boolean centre, equipped with the restricted operations, satisfies the equations of one standard axiomatisation of Nelson algebras [14]. An involutive distributive lattice \( \mathcal{N}(A) = A \times A \) is constructed together with an injective map \( \iota(x) = (x,x^\Delta) \). The embedding preserves joins globally and preserves meets on \( B(A) \). Prime-spectrum and prime-filter representation results are obtained for \( B \)-algebras. Under an injectivity condition on the pseudo-supplement map, an ultrafilter representation follows. The bi-implication \( x \leftrightarrow y = (x \Rightarrow y) \wedge (y \Rightarrow x) \) characterises equality and satisfies a transitivity inequality.

Abstract: Near-Heyting algebras generalise Heyting algebras to distributive nearlattices, linking each principal upset’s Heyting structure by a global identity. Sectional Hilbert nearlattices [23] drop this identity and instead build a filter and congruence theory from sectional residuation alone. We extend this programme to the Epstein–Horn \( B \)-algebra setting [11,24]. A sectional semi-\( B \)-nearlattice (SSBN-algebra) is a distributive nearlattice \( (A,\vee,1) \) whose implication \( \Rightarrow \) restricts, on every principal upset \( [a) \), to a semi-\( B \)-algebra satisfying the Birkhoff-center adjunction \( x \wedge_a b \leq z \Longleftrightarrow b \leq x \Rightarrow z \). SSBN-algebras sit in a hierarchy \( \mathbf{N_sB} \subseteq \mathbf{SSBN} \subset \mathbf{DNSB} \) between near-semi-\( B \)-algebras and the strictly larger class of distributive nearlattice semi-\( B \)-algebras. Within this hierarchy, ordinary (non-sectional) semi-\( B \)-algebras and near-semi-\( B \)-algebras turn out to be incomparable classes, with neither containing the other. For downward-directed SSBN-algebras, we build a complete lattice of implicative semi-\( B \)-filters and characterise generated filters through a sectional meet; this repairs a natural implication-based formula that fails when the adjoined element is not a Birkhoff element. From this characterisation we obtain an unconditional prime extension theorem, a separation theorem, a prime-spectrum embedding, and a congruence-and-quotient theory for the semi-\( B \)-algebra reduct.