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Equational bases for p-compatible identities

Penner P., Wismath S. L.

Demonstratio Mathematica

2010

Vol. 43, nr 3

511-518

A number of syntactical properties of identities, such as regularity, nor- mality, k-normality, externality and P-compatibility of identities have been extensively studied. We develop here a technique for producing from a basis for a variety V with certain idempotent terms a basis for the variety P(V), the smallest P-compatible variety to contain V. When V is finitely based, so is P(V).

Equational bases for k-normal identities

Penner P., Wismath S.

Demonstratio Mathematica

2008

Vol. 41, nr 4

733-742

The depth of a term may be used as a measurement of complexity of identities. For any natural number [...] have depth at least k. For any variety V, the k-normalization of V is the variety Nk(V) defined by all k-normal identities of V. We describe a process to produce from a basis for V a basis for Nk(V), for any variety V which has an idempotent term; when the type of V is finite and V is finitely based, this results in a finite basis for Nk(V) as well. This process encompasses several known examples, for varieties of bands and lattices, and allows us to give a new basis for the normalization of the variety PL of pseudo-complemented lattices.