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Free algebras on a fixed set. (English) Zbl 0788.08006
Free algebras on a set \(X\) which are not generated by the set \(X\) are considered (example: free group on a set \(X\), considered as an algebra with the operation of multiplication only). Define a relation \(F_ X\) on similar algebras by: \(AF_ X B\) if \(A\) is free on the set \(X\) in the class \(\{A,B\}\). It is shown that \(F_ X\) is a partial ordering. An operation \(h: A^ n\to A\) is called improper if \(h(a_ 1,\dots,a_ n)\varphi= h(a_ 1\varphi,\dots,a_ n\varphi)\) for every homomorphism \(\varphi: A\to A\). If \(h\) is an \(n\)-ary improper operation on an algebra \(A\) and \(k\) an \(n\)-ary improper operation on a similar algebra \(B\), then the operations \(h\), \(k\) are in relation Hom if \(h(a_ 1,\dots,a_ n)\varphi= k(a_ 1\varphi,\dots,a_ n\varphi)\) for every \(a_ 1,\dots,a_ n\in A\) and every homomorphism \(\varphi: A\to B\). If \(F(X)F_ X A\), where the set \(X\) is infinite and \(h\) is an \(n\)-ary improper operation on \(F(X)\), then there exists an \(n\)-ary improper operation \(k\) on \(A\) such that \(h\text{ Hom }k\).
Reviewer: J.Henno (Tallinn)
MSC:
08B20 Free algebras
08A40 Operations and polynomials in algebraic structures, primal algebras
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References:
[1] G. Grätzer: Universal Algebra. London, 1968. · Zbl 0182.34201
[2] I. Žembery: Almost equational classes of algebras. Algebra Universalis 23 (1986), 293-307. · Zbl 0621.08006
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