Boolean-Like Algebras of Finite Dimension: From Boolean Products to Semiring Products

We continue the investigation, initiated in Salibra et al. (Found Sci, 2020), of Boolean-like algebras of dimension n ( \(n\textrm{BA}\) s), algebras having n constants \(\mathsf e_1,\dots ,\mathsf e_n\) , and an \((n+1)\) -ary operation q (a “generalised if-then-else”) that induces a decomposition of the algebra into n factors through the so-called n-central elements. Varieties of \(n\textrm{BA}\) s share many remarkable properties with the variety of Boolean algebras and with primal varieties. Putting to good use the concept of a central element, we extend the Boolean power construction to that of a semiring power and we prove two representation theorems: (i) Any pure \(n\textrm{BA}\) is isomorphic to the algebra of n-central elements of a Boolean vector space; (ii) Any member of a variety of nBAs with one generator is isomorphic to a Boolean power of this generator. This yields a new proof of Foster’s theorem on primal varieties.