## A note on varieties with distributive subalgebra lattices.(English)Zbl 0777.08001

The author proves the following Theorem: For a variety $$V$$ of universal algebras, the following conditions are equivalent: 1) The lattice of subalgebras of every algebra in $$V$$ is distributive, 2) The variety $$V$$ is Hamiltonian and for every term $$p(x_ 1,x_ 2,x_ 3,x_ 4)$$ there are a term $$q(x_ 1,x_ 2,x_ 3,x_ 4)$$ and unary terms $$r_ i$$, $$s_ i$$, $$i=1,2,3,4$$ such that $$V$$ satisfies the identities $$p(x_ 1,x_ 2,x_ 3,x_ 4)=q(s_ 1(x_ 1),s_ 2(x_ 2),s_ 3(x_ 3),s_ 4(x_ 4))$$, $$r_ i(p(x_ 1,x_ 2,x_ 3,x_ 4))=s_ i(x_ i)$$, $$i=1,2,3,4$$.

### MSC:

 08A30 Subalgebras, congruence relations 08B05 Equational logic, Mal’tsev conditions

### Keywords:

Hamiltonian variety; lattice of subalgebras
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### References:

 [1] T. Evans B. Ganter: Varieties with modular subalgebra lattices. Bull. Austral. Math. Soc. 28 (1983), 247-254. · Zbl 0545.08010 [2] L. Klukovits: Hamiltonian varieties of universal algebra. Acta Sci. Math.(Szeged), 37 (1975), 11-15. · Zbl 0285.08004 [3] P.M. Winkler: Polynomial hyperforms. Algebra Univ. 17, (1983), 101-109. · Zbl 0558.08003
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