The minimal extension of sequences. II: On problem 17 of Grätzer and Kisielewicz
[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 \(A\in\{L_2,N_2,A_4\}\) are unknown. The proof of Theorem 1 implies Theorem 2. It states that the sequence \((0,1,2,9,114,\ldots,p_n(L_2),\ldots)\) is the minimal extension of the sequence \((0,1,2)\) in the class of all algebras from \(C(2,2)\) satisfying a non-regular identity.
scientific article; zbMATH DE number 1225616