Pure Hilbert Algebras with Infimum
In [6], iH-algebras were introduced in order to indicate an equational version of the class of Hilbert algebras where each pair of elements has infimum. These authors also proved that this variety hasthe class of Curry's implicative semilattices ([9]) as a proper subvariety. On the other hand, in [4] a special class of Hilbert algebras associated with ordered sets, which they called order algebras, were investigated. These algebras were also studied in [1] under the name of pure Hilbert algebras.Bearing in mind the above results, in this paper we introduce the notion of pure Hilbert algebras with infimum (or ipH-algebras, for short). Furthermore, we characterize the lattice of ipH-congruences and we determine the subdirectly irreducible ipH-algebras. Besides, we prove that subdirectly irreducible ipH-algebras are alsosubdirectly irreducible iH-algebras.
Congruences and Idealas in Hilbert Algebras
The concept of ideal for Hilbert algebras is introduced. It is presented a construction of a congruence having given ideal as its class. Moreover, it is shown that every congruence on a Hilbert algebra has this form.
Counting Finite Algebras in the Post Varieties
The G -spectrum of a class of algebras is the function that counts, up to isomorphism, the number of at most k -generated algebras in the class, for k =1,2,… . We consider the G -spectra of the varieties of algebras generated by a single two-element algebra. We show that such spectra are either polynomial, exponential, or at least doubly exponential as a function of k . For the varieties that have polynomial or exponential G -spectra we provide an exact formula for the function.