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

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
Full Text:

### References:

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