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...es Gn. A variety of algebras is called locally finite if and only if each of its finitely generated members is finite. A variety that is generated by a finite set of finite algebras is locally finite =-=[2]-=- . Theorem 3.6 (i) CS 0 is defined by x ω+2 = x 2 , (xy) ω+1 x = xyx, xyx (zx) ω = x (zx) ω yx. (6) (ii) For any n ≥ 1, CS 0 (Gn) is defined by Proof. x n+2 = x 2 , (xy) n+1 x = xyx, xyx (zx) n = x (z...

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...-simple if it does not have ideals except itself and possibly {0}. Every finite semigroup may be obtained from 0-simple semigroups by a sequence of ideal extensions. The classic theorem of Sushkevich =-=[3]-=- (which was arguably the first theorem in the algebraic theory of semigroups) shows that finite 0simple semigroups have the following structure. Let G be a group, let L and R be two sets and let P = (...

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...) . For varieties of semigroups U and V define U ∗ V to be the variety generated by {U ∗ V ; U ∈ U and V ∈ V}. Of course U wr V ∈ U ∗ V; in fact U ∗ V is generated by U wr V for all U ∈ U, V ∈ V (see =-=[1]-=-). We also use the analogous notions for pseudovarieties. Theorem 3.15 CS 0 (H) = SG(H) ∗ RZ for any variety of groups H. For any pseudovariety H of groups, CS 0 (H) = SG(H) ∗ RZ Proof. Notice that SG...

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...orphic images and finite direct products. There exists an important connection between the uniform word problem in a pseudovariety and finite partial algebras. This connection was found by Evans (see =-=[5]-=- or [10; Connection 2.2]). Recall that a partial universal algebra is a set with partial operations. If A is a partial universal algebra, B is a universal algebra of the same type, A ⊆ B and every ope...

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...mages then we obtain a decidable collection of finite semigroups. This collection is closed under subsemigroups, finite direct products and homomorphisms so it is a pseudovariety of finite semigroups =-=[4]-=- generated by the set of all finite 0-simple semigroups. Let V be a pseudovariety of groups. Define CS 0 (V) and B (V) to be the pseudovarieties generated respectively by finite 0-simple semigroups ov...

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...lian groups and let ZN2A be the class of all groups G such that the factor of G over its centre belongs to N2A. Let X be an arbitrary variety of groups such that ZN2A ⊆ X. Theorem 1.4 (Kharlampovich, =-=[8]-=-, [9]). The uniform word problem is undecidable for the following classes of groups: G; N; G ∩ N; G ∩ X; N ∩ X; N ∩ X ∩ G. 3the sets is undecidable. On the other hand the universal word problem is so...

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...set of subsemigroups of direct products of (finite) completely 0-simple semigroups is also not recursive. Kublanovsky uses the unsolvability of the uniform word problem for finite groups (Slobodskoii =-=[14]-=-). Recall that if K is a class of universal algebras of some type then the uniform word problem for this class asks whether there exists an algorithm which, given a system of relations ui = vi, i = 1,...

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... S↕ ∗ RZ also has this basis whence CS 0 (T ) = S↕ ∗ RZ (in [1], S↕ is called Com11 and RZ is called D1). Since C2 ∈ CS 0 (T ) ⊆ Sl ∗ RZ then CS 0 (T) ⊆ Sl ∗ RZ; that is, CS 0 (T) = Sl ∗ RZ. Since by =-=[7]-=-, H∗RZ = CS(H) then by Lemma 3.13 CS 0 (H) = SG(H) ∗ RZ. In the corresponding proof for pseudovarieties, note that by [1; Corollary 10.6.8], CS(H) = H ∗ RZ. We get CS 0 (H) = SG(H) ∗ RZ. Remark 3.16 T...

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...groups and let ZN2A be the class of all groups G such that the factor of G over its centre belongs to N2A. Let X be an arbitrary variety of groups such that ZN2A ⊆ X. Theorem 1.4 (Kharlampovich, [8], =-=[9]-=-). The uniform word problem is undecidable for the following classes of groups: G; N; G ∩ N; G ∩ X; N ∩ X; N ∩ X ∩ G. 3the sets is undecidable. On the other hand the universal word problem is solvabl...

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... information is known about pseudovarieties of groups with undecidable uniform word problem (see [10]). The undecidability of the uniform word problem in the class of all groups was proved by Novikov =-=[12]-=-. The undecidability of the uniform word problem in the class of finite groups has been proved by Slobodskoii [14]. The following results of Kharlampovich imply that many other pseudovarieties of grou...