Józef Dudek

Idempotent algebras with log-linear free spectra

In this paper we consider idempotent algebras whosepn-sequences (the sequences of the numberspn of essentiallyn-ary polynomials) have a subexponential rate of growth. Studying the symmetry groups of polynomials we establish some consequences of this property. In particular, a new characterization of semilattices is obtained, and all idempotent commutative algebras with log-linear free spectra are described.

On finite models of regular identities.

It is a known result of Austin that there exist nonregular identities with all nontrivial models being infinite. In this note a certain analogue of this result for regular identities is presented and some remarks in this connection are given

Small idempotent clones I

G. Grätzer and A. Kisielewicz devoted one section of their survey paper concerning pn-sequences and free spectra of algebras to the topic “Small idempotent clones” (see Section 6 of [18]). Many authors, e.g., [8], [14, 15], [22], [25] and [29, 30] were interested in pn-sequences of idempotent algebras with small rates of growth. In this paper we continue this topic and characterize all idempotent groupoids (G, ·) with p2(G, ·) ≤ 2 (see Section 7). Such groupoids appear in many papers see, e.g. [1], [4], [21], [26, 27], [25], [28, 30, 31, 32] and [34].

The minimal extension of the sequence 〈, 0, 0,3〉

The main theorem proved in this paper is that the sequence ∮, 0, 3 has the unique minimal extension to the sequence ∮, 0, 3,..., 2n -1,... and that any algebra having this as itspn-sequence is equivalent to an algebra in one of two explicitly described equational classes.