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.