Modular median algebras generated by some partial modular median algebras. (English) Zbl 0889.08010

Draškovičová, Hilda

Math. Slovaca 46, No. 4, 405-412 (1996).

Let \(\mathcal {M}\) denote the variety of algebras with one ternary operation \((abc)\) satisfying the identities \((abb) = b\) and \(((abc)dc) = (ac(dcb))\). The subvariety \(\mathcal {T}\) of the variety \(\mathcal {M}\) is given by the identity \(((abc)de) =((ade)(bde)(cde))\). It is known that the lattice of subvarieties of the variety \(\mathcal {T}\) forms a strictly increasing sequence (a chain) of varieties \(\mathcal {T}_{n}\), \(n = 1,2,\dots ,\omega \), and \(\mathcal {T} = \mathcal {T}_{\omega }\). For each \(\mathcal {T}_{n}\), \(1< n< \omega \), a finite base of identities is given. The free algebra \(F_{\mathcal {M}} (3)\) on three generators over the variety \(\mathcal {M}\) belongs to the variety \(\mathcal {T}\). Since there is not known anything about the free algebra \(F_{\mathcal {M}} (4)\) on four generators over \(\mathcal {M}\), results about the algebras in \(\mathcal {M}\) or in \(\mathcal {T}\), respectively, which are generated by some partial algebras are given.

MSC:

08B15

Lattices of varieties

Keywords:

modular median algebra; free algebra; partial algebras

References:

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