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Identities for globals (complex algebras) of algebras. (English) Zbl 0668.08005

The global \({\mathfrak P}({\mathfrak A})\) of an algebra \({\mathfrak A}=<A,F>\) is the family of all nonvoid subsets of A with operations given by \(f(A_ 1,...,A_ n)=\{f(a_ 1,...,a_ n)\); \(a_ i\in A_ i\), \(1\leq i\leq n\}\) for any \(f\in F\). \({\mathfrak P}_ 0({\mathfrak A})\) denotes the analogous algebraic structure on the set of all subsets of A, including the empty set. For a variety V, the symbol \({\mathfrak P}(V)\) (\({\mathfrak P}_ 0(V))\) denotes the variety determined by the class \(\{\) \({\mathfrak P}({\mathfrak A})\); \({\mathfrak A}\in V\}\) (\(\{\) \({\mathfrak P}_ 0({\mathfrak A})\); \({\mathfrak A}\in V\}\), resp.). A term p is linear if no variable symbol occurs more than once in p. An identity \(p\equiv q\) is linear if both terms p and q are linear; \(p\equiv q\) is regular if \(var(p)=var(q).\)
For an arbitrary variety V of finitary algebras it is proved: (1) The identities satisfied by \({\mathfrak P}(V)\) are precisely those identities resulting through identification of variables from the linear identities true in V. (2) The identities satisfied by \({\mathfrak P}_ 0(V)\) are precisely those regular identities resulting through identification of variables from the linear identities true in V.
For varieties of lattices and of groups it is shown: (3) There is exactly one nontrivial variety determined by globals of lattices. (4) There are exactly three nontrivial varieties determined by globals of groups.
Reviewer: J.Duda

MSC:

08B05 Equational logic, Mal’tsev conditions
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