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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
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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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