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Kiss, The Shape of Congruence Lattices , pdf. Willard, An overview of modern universal algebra , pp. Andretta, K. Kearnes and D. Zambella, Lecture Notes in Logic, vol. Press, Freese and O. Articles and Notes J. II Math.
Characterization of protomodular varieties of universal algebras
Birkhoff, On the structure of abstract algebras , Proc. Cmabridge Phil.
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Freese, Computing congruences efficiently , Algebra Universalis, 59 , Freese, Notes on the Birkhoff Construction , pdf. Freese and M. Valeriote, On the complexity of some Maltsev conditions , Internation J. Algebra and Computation, to appear. The categories y,p and N y l R y are isomorphic.
The structures A and M A, z have the same polynomial functions and the same congruences. The variety of all quasigroups is a modular variety, in fact permutable. Following Jezek and Kepka , we call this subvariety the variety of cia-quasi- groups commutative, idempotent, abelian quasigroups—any quasigroup satisfying a is abelian. It is a straightforward verification that k is a congruence on Q X Q and that the diagonal is a congruence class. To define module addition and subtraction, choose any z g Q to be the zero element.
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This has been known since as a special case of "Toyoda's Theorem" . Similarly, the product in the ring corresponds to the product of rational numbers in F use induction on the length of a term. Thus we see that the ring R y — Z[i]. We would now like to apply Theorem 3. The key is to show that the information contained in the module N y is not needed. This will follow from Theorem 3.
Let y be an abelian variety of finite type. Then yis deductive if and only if y has the subalgebra condition and R y is deductive. Let A be any non tri vial member of y and let k be the congruence on A X A that contains the diagonal as a class. Thus by Theorem 3. Let SI be a subquasivariety of y.
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It is a somewhat tedious, but not difficult, exercise to show that SM is closed under the formation of submodules, products and ultraproducts use the corresponding property of J together with 3. Furthermore, S is closed under homomorphic images if and only if SlMis so closed. D Returning to cia-quasigroups, let us apply Theorem 3. The ring is not Artinian, but as remarked earlier, Example 2.
On the other hand, Z[j] is a PID and, by 2. Translating back to quasigroups, every proper subvariety of cia-quasi- groups is a deductive variety. We now apply Theorems 2. As an example: for any natural number n, the variety of abelian groups of exponent n is directly representable. We require the following properties of directly representable varieties from McKenzie [14, Theorems 5. Then yis permutable hence modular and every subdirectly irreducible algebra in yis finite and either simple or abelian.
Thus we have the following characterization. Let y be a directly representable variety of finite type, yis deductive if and only if: 1 y is equationally complete, or 2 y has the subalgebra condition and yah is deductive. Let y be deductive.
Then by 3. Thus y has no nontrivial abelian algebras, so y must be semisimple. By Corollary 3. On the other hand, if y has the subalgebra condition, then alternative 2 obviously holds. For the converse, let y be equationally complete. Let M be a finite, nontrivial algebra of y of smallest cardinality. M is simple, has no proper, nontrivial subalgebras and generates y, which is permutable.
Therefore by [16, Theorems 1. Since M can be embedded into every direct power of M, and since V is locally finite, M is primitive. Therefore by Theorem 3. Now suppose y has the subalgebra condition and yah is deductive. By Theorem 3. There are two cases. Case 1. B is abelian. Then 6 d [1,1] in Con A. By assumption, yah is deductive, so B g IS C by 3. Case 2. B is nonabelian. Then B is simple, so 0 is a coatom of Con A. D Example. Thus y is not deductive. Since the order of Q is prime, Q is simple.
A is isomorphic to the group Z2, so A is simple and abelian. By [3, Theorem 1. Q is nonabelian since it is simple, but has a subalgebra and y has the subalgebra condition since Q and A have idempotents. The subvariety yah is generated by A, so the associated ring is Z2.
Related Universal Algebras
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