Representable pseudo-interior algebras (Q1966159)

The notion of pseudo-interior algebra was introduced by \textit{W. J. Blok} and \textit{Don Pigozzi} [Algebra Univers. 31, 1-35 (1994; Zbl 0817.08005)]. They represent the logic inherent in varieties with a commutative, regular ternary deductive term. An algebra \(A=\langle A,\cdot ,\rightarrow ,1,{}^0\rangle \) of type (2,2,0,1) is called a pseudo-interior algebra if: (1) \(\langle A,\cdot ,1\rangle \) is a semigroup with the left identity 1. (2) \({}^0\) is a pseudo-interior operation on \(\langle A,\cdot ,1\rangle \), i.e.\ it satisfies the identities \[ x\cdot y = x^0 \cdot y, \;x^0 \cdot x = x^0, \;x^0 \cdot y^0 = y^0 \cdot x^0, \;1^0 = 1. \] (3) \(\rightarrow \) is the open left-residuation on \(\langle A,\cdot ,1\rangle \), i.e.\ for all \(x\) and \(y\) in \(A\) the element \(x\rightarrow y\) is open, and \(\forall \;u\): \((u\cdot x \leq _r y \Leftrightarrow u^0 \leq _r x\rightarrow y)\), where \(x\leq _r y\) iff \((\exists \;u : x = u\cdot y)\). For a given topological space \(\langle X,T \rangle \), the powerset \(\mathcal P(X)\) with the ordinary unary interior operation \({}^0\), the binary operation \(x\cdot y = x^0 \cap y\) is a semigroup with the left identity \(X\) and a pseudo-interior operation. For \(x,y \in \mathcal P(X)\), \(x\leq _r y\) iff \(x\) is a relatively open subset of \(y\). The set \(\{u\in T: u\cap x \leq _r y\}\) has the greatest element denoted by \(x\rightarrow y\). A pseudo-interior algebra of this form is called a topological one. The main theorem of the paper says that every pseudo-interior algebra is isomorphic to the subalgebra of a topological one. This gives a solution to a problem posed by Blok and Pigozzi.