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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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