We show that a variety with J´onsson terms t1, . . . , tn´1 has directed J´onsson terms d1, . . . , dn´1 for the same value of the indices, solving a problem raised by Kazda et al. Refined results are obtained for locally finite varieties.

Some cardinal characteristics related to the covering number and the uniformity of the meagre ideal

We define a procedure for translating a given first-order formula to an equivalent modal formula, if one exists, by using tableau-based bisimulation invariance test. A previously developed tableau procedure tests bisimulation invariance of a given first-order formula, and therefore tests whether that formula is equivalent to the standard translation of some modal formula. Using a closed tableau as the starting point, we show how an equivalent modal formula can be effectively obtained.

Given the congruence lattice L of a finite algebra A that generates a congruence permutable variety, we look for those sequences of operations on L that have the properties of higher commutator operations of expansions of A. If we introduce the order of such sequences in the natural way the question is whether exists or not the largest one. The answer is positive. We provide a description of the largest element and as a consequence we obtain that the sequences form a complete lattice.

A b s t r a c t. We investigate the complexity of the standard translation of lambda calculus into combinatory logic. The main result shows that the asymptotic growth rate of the size of a translated term is (n 3 ) in worst-case, where n denotes the size of the lambda term.

A b s t r a c t. We will introduce the special kind of the order relations into recursively enumerable sets and prove that they can be used to distinguish (albeit in a non-constructive way) between recursive and non-recursive sets.

In [3] we have claimed that finite Heyting algebras with successor only generate a proper subvariety of that of all Heyting algebras with successor, and in particular all finite chains generate a proper subvariety of the latter.As Xavier Caicedo made us notice, this claim is not true.He proved, using techniques of Kripke models, that the intuitionistic calculus with S has finite model property and from this result he concluded that the variety of Heyting algebras with successor is generated by its finite members [2].This fact particularly affects Section 3.2 of our article.Concretely, in Remark 3.3, our claim "Let K be a class of S-Heyting algebras of height less or equal to a fixed ordinal .Using the categorical duality between S-Heyting algebras and S-Heyting spaces, it can be shown that the elements of classes H(K), S(K) and P(K) have also height less or equal to .Here H, S and P are the class operators of universal algebra.Hence for each ordinal , the class of S-Heyting algebras of height less or equal to is a variety" is not true as stated.It remains valid only if is a finite ordinal.

Formulas for computing the number of Df2-algebra structures that can be defined over Bn, where Bn is the Boolean algebra with n atoms, as well as the fine spectrum of Df2 are obtained. Properties of the lattice of all subvarieties of Df2, (Df 2), are exhibited. In particular, the poset Sifin(Df2) is described.

This paper studies some properties of the so-called semilattice-based logics (which are dened in a standard way using only the order relation from a variety of algebras that have a semilattice reduct with maximum) under the assumption that its companion assertional logic (dened from the same variety of algebras using the top element as representing truth) is algebraizable. This describes a very common situation, and the conclusion of the paper is that these semilattice-based logics exhibit some of the good behaviour of protoalgebraic logics, without being necessarily so. The main result is that all these logics have enough Leibniz lters, a fact previously known in the literature to occur only for protoalgebraic logics. Another signicant result is that the two companion logics coincide if and only if one of them enjoys the characteristic property of the other, that is, if and only if the semilattice-based logic is algebraizable, and if and only if its assertional companion is selfextensional. When these conditions are met, then the (unique) logic is nitely, regularly and strongly algebraizable and fully Fregean; this places it at some of the highest ranks in both the Leibniz hierarchy and the Frege hierarchy. Let K be a class of algebras of the same (arbitrary) similarity type. The class K is said to be semilattice-based when each algebra in K has a semilattice reduct uniformly dened