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Essential variables and positions in terms. (English) Zbl 1197.08003

Let \(\Sigma\) be a set of identities of a given type \(\tau\). A \(D\)-closure of \(\Sigma\) is defined as the smallest set \(D(\Sigma)\) containing \(\Sigma\) and satisfying five “deductive rules”. By [S. Burris and H. P. Sankappanavar, A course in universal algebra. Millennium edition. http://www.math.uwaterloo.ca/~snburris/htdocs/ualg (2000)], \(\Sigma\models t\approx s\) if and only if \(t\approx s\in D(\Sigma)\), and \(D(\Sigma)\) is a fully invariant congruence. The author investigates properties equivalent to some deductive rules in which notions of essential variables and positions are applied. If \(\Sigma\) satisfies “deductive rules” together with “\(\Sigma\) replacement”, but without “term positional replacement”, then \(\Sigma\) is called \(\Sigma R\)-deductively closed. The \(\Sigma R\)-closure of \(\Sigma\) is defined naturally. One of the main results is the completeness theorem for \(\Sigma R\)-equational logic. Further, it is shown that the closure \(\Sigma R(\Sigma)\) is a fully invariant congruence, but in general \(\Sigma R(\Sigma)\neq D(\Sigma)\). In the last part, \(\Sigma\)-balanced identities are studied and the following theorem is proved: Let \(\Sigma\subset\) Id\((\tau)\) be a set of \(\Sigma\)-balanced identities. If there is a \(\Sigma R\)-deduction of \(t\approx s\) with \(\Sigma\)-balanced identities then \(t\approx s\) is a \(\Sigma\)-balanced identity of type \(\tau\).

MSC:

08B05 Equational logic, Mal’tsev conditions
03C05 Equational classes, universal algebra in model theory
08A02 Relational systems, laws of composition
08B15 Lattices of varieties
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