The minimal extension of sequences. II: On problem 17 of Grätzer and Kisielewicz
type: Item
instance of: scholarly article
title: The minimal extension of sequences. II: On problem 17 of Grätzer and Kisielewicz
zbMATH Open document ID: 0902.08006
DOI: 10.1023/A:1004363119941
author: Józef Dudek
published in: Periodica Mathematica Hungarica
publication date: 1998-11-22T00:00:00Z
review text: [For Part I and III see Zbl 0627.08001 and Zbl 0865.08001, respectively.] For an algebra \(A\), let \(p_n(A)\) be the number of all essentially \(n\)-ary term operations of \(A\). An infinite or finite sequence \((a_0,\ldots)\) of cardinals is called \textit{representable} (in a given class of algebras) if there exists an algebra \(B\) (belonging to this class) such that \(p_n(B)= a_n\) for all \(n\). A sequence \(a^*=(a_0,\ldots,a_m,\ldots)\) of cardinals is the \textit{minimal extension} of the sequence \(a=(a_0,\ldots,a_m)\) (in a class \(K\) of algebras) if \(a^*\) is representable (in \(K\)) and for every algebra \(A\) (\(\in K\)) which represents \(A\) we have \(p_n(A)\geq a_n\) for all \(n\). The author deals with the following Problem 17 of \textit{G. Grätzer} and \textit{A. Kisielewicz} [in: A. Romanowska et al. (eds.), Universal Algebra and Quasigroup Theory, Lect. Conf., Jadwisin/Pol. 1989, Res. Expo. Math. 19, 57-88 (1992; Zbl 0772.08001)]: does the sequence \((0,1,2)\) have the minimal extension property with respect to the class \(C(2,2)\) of all commutative algebras of type \((2,2)\)? To formulate the results of the article, we need some notation. Let \(L_2\) be the 2-element lattice. By \(N_2\) the author denotes the algebra \(\langle\{-1,0, 1,2\};\;\circ\rangle\) where \(x\circ y=x\) if \(x=y\); \(x+y=1+\max(x,y)\) if \(x\neq y\) and \(x,y\leq 1\); and \(x+y=2\) otherwise. Finally, let be \(A_4=\langle\{0,a,b, 1\};\;+,\cdot\rangle\) where \(+\) and \(\cdot\) are binary operations such that, for all \(x,y\in A\), a) \(x+x=x\), \(x+y=y+x\), \(0+x=0\), \(a+b=a+1=1\) and \(b+1=1\); b) \(x\cdot x=x\), \(x\cdot y=y\cdot x\), \(0\cdot x=0\), \(a\cdot b=0\) and \(a\cdot 1=b\cdot 1=1\). The main result of the article is Theorem 1. It states that if \(A\in C(2,2)\) and \(A\) represents the sequence \((0,1,2)\), then \(A\) contains a subalgebra isomorphic to one of the algebras \(L_2\), \(N_2\) and \(A_4\). The numbers \(p_n (A)\) where \(A\in\{L_2,N_2,A_4\}\)…
Mathematics Subject Classification ID: 08A40
zbMATH DE Number: 1225616
zbMATH Keywords: term operation
zbMATH Keywords: \(p_n\)-sequence
zbMATH Keywords: lattice
zbMATH Keywords: groupoid
zbMATH Keywords: commutative algebra
zbMATH Keywords: minimal extension property
reviewed by: Boris Vernikov
MaRDI profile type: Publication
full work available at URL: a:1004363119941
OpenAlex ID: W1692017151
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description: scientific article; zbMATH DE number 1225616