Online research in philosophy

Let be a finite collection of finite algebras of finite signature such that SP( ) has meet semi-distributive congruence lattices. We prove that there exists a finite collection 1 of finite algebras of the same signature, , such that SP( 1) is finitely axiomatizable.We show also that if , then SP( 1) is finitely axiomatizable. We offer new proofs of two important finite basis theorems of D. Pigozzi and R. Willard. Our actual results are somewhat more general than this abstract (...) indicates. (shrink)

Assume that all algebras are atomless. (1) $Spind(A x B) = Spind(A) \cup Spind(B)$ . (2) $(\prod_{i\inI}^{w} = {\omega} \cup \bigcup_{i\inI}$ $Spind(A_{i})$ . Now suppose that $\kappa$ and $\lambda$ are infinite cardinals, with $kappa$ uncountable and regular and with $\kappa \textless \lambda$ . (3) There is an atomless Boolean algebra A such that $\mathfrak{u}(A) = \kappa$ and $i(A) = \lambda$ . (4) If $\lambda$ is also regular, then there is an atomless Boolean algebra A such that $t(A) = \mathfrak{s}(A) = (...) \kappa$ and $\mathfrak{a}(A) = \lambda$ . All results are in ZFC, and answer some problems posed in Monk [01] and Monk [ $\infty$ ]. (shrink)

We exhibit a construction which produces for every Turing machine T with two halting states μ 0 and μ -1 , an algebra B(T) (finite and of finite type) with the property that the variety generated by B(T) is residually large if T halts in state μ -1 , while if T halts in state μ 0 then this variety is residually bounded by a finite cardinal.

We determine precisely those locally finite varieties of unary algebras of finite type which, when augmented by a ternary discriminator, generate a variety with a decidable theory.