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On Neat Reducts of Algebras of Logic

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Abstract

SC α, CA α, QA α and QEA α stand for the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasipolyadic algebras, and quasipolyadic equality algebras of dimension α, respectively. Generalizing a result of Németi on cylindric algebras, we show that for K ∈ {SC, CA, QA, QEA} and ordinals α < β, the class Nr α K β of α-dimensional neat reducts of β-dimensional K algebras, though closed under taking homomorphic images and products, is not closed under forming subalgebras (i.e. is not a variety) if and only if α > 1.

From this it easily follows that for 1 < α < β, the operation of forming α-neat reducts of algebras in K β does not commute with forming subalgebras, a notion to be made precise.

We give a contrasting result concerning Halmos' polyadic algebras (with and without equality). For such algebras, we show that the class of infinite dimensional neat reducts forms a variety.

We comment on the status of the property of neat reducts commuting with forming subalgebras for various reducts of polyadic algebras that are also expansions of cylindric-like algebras. We try to draw a borderline between reducts that have this property and reducts that do not.

Following research initiated by Pigozzi, we also emphasize the strong tie that links the (apparently non-related) property of neat reducts commuting with forming subalgebras with proving amalgamation results in cylindric-like algebras of relations. We show that, like amalgamation, neat reducts commuting with forming subalgebras is another algebraic expression of definability and, accordingly, is also strongly related to the well-known metalogical properties of Craig, Beth and Robinson in the corresponding logics.

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Authors and Affiliations

  1. Department Of Mathematics Faculty Of Science, Cairo University, Giza, Egypt

    Tarek Sayed Ahmed

  2. Mathematical Institute, The Hungarian Academy Of Science, Pf. 127, 1364, Budapest, Hungary

    Istvan Németi

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  1. Tarek Sayed Ahmed
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Ahmed, T.S., Németi, I. On Neat Reducts of Algebras of Logic. Studia Logica 68, 229–262 (2001). https://doi.org/10.1023/A:1012447223176

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