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- W80905092 abstract "In the sequel, H, S, P , Pfin shall denote the following operators on classes of algebras ( cf. [23]): H taking homomorphic images of algebras, S taking subalgebras, P taking products and Pfin for taking finite products of algebras (i.e. direct products of nonvoid finite families of algebras). Moreover, as usually: I is the operator of taking isomorphic copies of algebras, PS , (Pfin)S denote the operators of the formation subdirect products of nonvoid, finite families of algebras, respectively. PU will denote the operator of ultraproducts (cf. [14], [18]). First G. Birkhoff’s Theorem 1935, [2] asserts that a set Σ of identities of a given type τ : T → N can be represented in the form Σ = θ(K), i.e. is an equational theory if and only if Σ is closed under rules (1) − (5) of derivation (called Birkhoff’s rules of derivation), [14], [27]. Second Birkhoff’s Theorem [2] asserts that a nonempty class of algebras of a given type τ : T → N is a variety (i.e. is closed under operations of passing to subalgebras, homomorphic images and also arbitrary direct products) if and only if it is an equational theory." @default.
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- W80905092 title "Jónsson's lemma for regular and nilpotent shifts of pseudovarieties" @default.
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