Dudek, J. On the minimal extension of sequences. (English) Zbl 0627.08001 Algebra Univers. 23, 308-312 (1986). An infinite (or finite) sequence \(a_0,\ldots,a_n,\ldots\) of cardinals is representable in a class \(K\) of algebras if there exists an algebra \(A\) in \(K\) such that \(p_n(A)=a_n\) all \(n\). \(p_n(A)\) denotes the number of essentially \(n\)-ary terms over \(A\). \(a^*=<a_0,\ldots,a_m, \ldots>\) is a minimal extension of the sequence \(a=<a_0,\ldots,a_m>\) if \(a^*\) is in \(K\) and we have \(p_n(A)\geq a_n\) for all \(n\). The main result is that the sequence \(<0,0,2>\) has a minimal extension to the sequence \(<0,0,2,\ldots,p_n(A),\ldots>\) in the class of all commutative groupoids where \(A=<\{1,2,3,4\};\circ >\) is defined by \(x\circ y=x\) for \(x=y\) \(x\circ y=1+\max \{x,y\}\) for \(x\ne y\) and x,y\(\le 3\) and \(x\circ y=4\) otherwise. Reviewer: Dietmar Schweigert (Kaiserslautern) Cited in 2 ReviewsCited in 7 Documents MSC: 08A40 Operations and polynomials in algebraic structures, primal algebras 20M14 Commutative semigroups Keywords:representable sequence; essentially n-ary terms; minimal extension; commutative groupoids PDFBibTeX XMLCite \textit{J. Dudek}, Algebra Univers. 23, 308--312 (1986; Zbl 0627.08001) Full Text: DOI References: [1] W.Belousov,Foundations of the theory of quasigroups and loops, Moscow 1967 (in Russian). · Zbl 0229.20075 [2] J. Dudek,On binary polynomials in idempotent commutative groupoids, Fund. Math.120 (1984), 187-191. · Zbl 0555.20035 [3] G. Gr?tzer, Universal algebra, Springer-Verlag, Berlin-Heidelberg-New York 1979. [4] G. Gr?tzer,Composition of functions, Proceedings of the Conference on Universal aigebra, 1969, Queen’s University, Kingston, (1970), 1-106. [5] G. Gr?tzer andR. Padmanabhan,On idempotent, commutative and non-associative groupoids, Proc. Amer. Math. Soc.,28 (1971), 75-80. · Zbl 0215.34501 [6] A. Kisielewicz,Minimal extensions of minimal representable sequences, Algebra Universalis,22 (1986), 244-252. · Zbl 0605.08001 · doi:10.1007/BF01224030 [7] R. E. Park,A four element algebra whose identities are not finitely based, Algebra Universalis11 (1980), 225-260. · Zbl 0449.08005 · doi:10.1007/BF02483103 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.