English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

1. Given an algebra A, we denote by pn = pn(A) the number of essentially n-ary polynomials over A. For definitions we refer the reader to [5] and [6]. We say that an infinite (or finite) sequence (a0, . . .) of cardinals is representable if there exists an algebra A0 such that pn(A0) = an for all n. If additionally A0 is taken from a given class of algebras, then we say that the sequence is representable in that class. A sequence a∗ = (a0, . . . , am, . . .) of nonnegative integers (cardinals) is a minimal extension of the sequence a = (a0, . . . , am) (in a given class K of algebras) if a∗ is representable (in the class K) and for every algebra A (∈ K) which represents a = (a0, . . . , am) we have pn(A) ≥ an for all n. Problems concerning minimal extensions of sequences were raised by G. Grätzer in [4]. He also initiated (together with R. Padmanabhan and J. P lonka) the problems of characterization of algebras (varieties) by means of the number pn (see Problem 42, p. 195 of [5]). In this paper we deal with Problem 16 of G. Grätzer and A. Kisielewicz [6]: Does the sequence (0, 1, n) have a minimal extension, for any n ≥ 0? The main result was presented during the Conference on Logic and Algebra dedicated to Roberto Magari on his 60th birthday, Pontignano (Siena), 26–30 April 1994 (see Theorem 2 of [3]).

The minimal extension of sequences III. On problem 16 of Grätzer and Kisielewicz