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An elementary theory of imbeddability skeleta of discriminator varieties. (Russian) Zbl 0756.08003
Let \(\mathfrak M\) be a variety of algebras and \(\text{I}\mathfrak M\) be the set of isomorphic types (isotypes) of algebras from \(\mathfrak M\). For \(a\), \(b\) from \(\mathfrak M\), there holds \(a\leq b\) iff there exists an isomorphic embedding of an algebra of isotype \(a\) in an algebra of isotype \(b\). The quasi-ordered class \(\langle\text{I}{\mathfrak M},\leq\rangle\) is called an imbeddability skeleton of \(\mathfrak M\). The main result of this paper is the following: Theorem. If \(\mathfrak M\) is a variety containing a quasi-primal algebra without one-element subalgebras, then the elementary theory of the imbeddability skeleton of \(\mathfrak M\) (i.e. \(\langle\text{I}{\mathfrak M},\leq\rangle\)) is undecidable.

MSC:
08B05 Equational logic, Mal’tsev conditions
03D35 Undecidability and degrees of sets of sentences
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