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.