By B. Csákány, J. Schmidt

ISBN-10: 0720407257

ISBN-13: 9780720407259

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3] A. D a y , p-modularity implies modularity in equational classes, Alg. , 3 (1973), 398-399. [4] A . D a y , Lattice conditions implying congruence modularity, Alg. , in print. [5] A. D a y , Splitting lattices generate all lattices, Alg. , in print. [6] B . J o n s s o n , Identities in congruence varieties, preprint. [7] P . M e d e r l y , Three Mal'cev type theorems and their applications, Mat. , 25 (1975), 83-85. [8] R. M c K e n z i e , Equational bases and non-modular lattice varieties, Trans.

Zf) * ρ{ζχ,. . , zr))) and assume furthermore (. . (gx o g2) o . . g^_ χ) = 1. method demonstrated above we get 1 = p ( z p . . , z r ) * i ( Z p . . ,z f ) 1 = qfizj,. . , z r ) » p i z j , . . ,zr) = q(z1,... ,zr). 2) a(b * b) * a= (b * b) * (a * a) = 1. It remains to guarantee a o (b o c)= (a o b) o c a o (a * b) = (b A a)a (b2 Ab)* (b2 Ab)a = a and (a o b) A b = a. 2). 11). Hence we have proved the main result of this paper: Theorem 1. Let (S, ·, *, Δ, o) be an algebra in which the equation (F)with f defined as above is true.

There are short one-identity axioms for the Boolean algebra which are based on the type of equation (B). 3. /-GROUP-CONE-IDENTITIES Theorem 3. (F) is a one-identity axiom of the write · instead of o, : instead of A and - 50 - l-group-cone if we / , = ( · · · ( ? ° g2)° ■••° 8η) with gx :=x1 *(xQ *x0) g2 := x2 * ((x2lx22 * * 2 3 ) * (x22 * (x2l * x23))) g3 := x 3 * ((jf32 · (x 3 1 * J C 3 3 ) ) . — ΛΓγ * v ^ ^ v ^ i * ^72^ * ^ 7 1 "^72^72^* Proof. By gj we have a * a = b * b =:l9 a* 1 = 1. By g j , g3 we get (i) ab * c = b * (a * c).

### Contributions to Universal Algebra by B. Csákány, J. Schmidt

by Steven

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