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On the minimal extension of sequences. (English) Zbl 0627.08001

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.

MSC:

08A40 Operations and polynomials in algebraic structures, primal algebras
20M14 Commutative semigroups
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References:

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