Reflection-closed varieties of multisorted algebras and minor identities. (English) Zbl 1412.08003

The notion of reflection conceived by L. Barto et al. [Isr. J. Math. 223, 363–398 (2018; Zbl 1397.08002)] is generalized to multisorted algebras in the sense of W. Wechler [Universal algebra for computer scientists. Berlin etc.: Springer-Verlag (1992; Zbl 0748.68002)]. Minor identities have only one function symbol on either side; they generalize the height-1 identities of Barto et al. [loc. cit.]. It is shown that a class of similar multisorted algebras is definable by a set of minor identities iff it is closed with respect to reflection and isomorphic copies of direct products. Such classes are dubbed “minor varieties”. The associated notion of “minor-equational theory” can be characterized through explicit, albeit somewhat intricate closure conditions.


08A68 Heterogeneous algebras
08B15 Lattices of varieties
03C05 Equational classes, universal algebra in model theory
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