Polyadic and cylindric algebras of sentences

Abstract

In this note we give an interpretation of cylindric algebras as algebras of sentences (rather than formulas) of first order logic. We show that the isomorphism types of such algebras of sentences coincide with the class of neat reducts of cylindric algebras. Also we show how this interpretation sheds light on some recent results. This is done by likening Henkin's Neat Embedding Theorem to his celebrated completeness proof.Cylindric and quasipolyadic algebras of formulas are oriented to languages without operation symbols, see [21, Section 4.3]. All terms other than variables are ignored. Though terms may be treated algebraically afterwards, see for example [21,19,20], there seems to be some artificiality in this approach. While constructing the algebra one omits terms, and then tries to invent means of extracting terms from the algebra originally constructed without them. There have been attempts to circumvent this drawback. Indeed, algebraizing first order logic by allowing terms other than variables, i. e. treating them seriously from the very beginning, has led to the more general notion of supercylindric and superpolyadic algebras [16]. Other work in this direction (dealing with cylindric algebras of formulas with terms) includes [17]. In what follows, we also allow terms other than variables, namely individual constants, but nothing else. However, we will not be concerned with (super)cylindric nor (super)polyadic algebras of formulas with terms. Instead, we introduce cylindric algebras and quasipolyadic algebras of sentences. The novelty -to the approach adopted in the monograph [21, Section 4.3] -occurring here is twofold: Firstly, considering languages endowed with individual constants, i. e. terms other than variables. Secondly, considering cylindric and polyadic algebras of sentences and not of formulas. The trick is that we allow constants in our languages. In more detail: Let α be an ordinal. Let L be an ordinary first order language with two propositional constants and ⊥, and a list c k of individual constants of order type α. L has no operation symbols of rank > 0, but as usual, the list of variables is of order type ω. Denote by Sn L the set of all L-sentences. For ϕ ∈ Sn L and k < α let

On amalgamation of reducts of polyadic algebras

“…in algebraic logic. 9 At least one such transformation should be non-surjective for else the Sain-Thompson nonfinite axiomatizability result applies. In other words if G consists only of surjective transformations only, then the class of concrete G algebras is not finite-schema axiomatizable.…”

“…In the next two remarks we elaborate on representing G algebras as algebras of sentences in first order languages endowed with individual constants which, to the best of our knowledge, is a novel interpretation of such algebras originating with Amer [9]. The number of individual constants determines the dimension of the algebra in question.…”

Some results about neat reducts

“…Furthermore, for such algebras, neat reducts commute with forming subalgebras; hence, this class has SUPAP [18]. In [26] the NET of Henkin is likened to his completeness proof; therefore, it is not a coincidence that interpolation results and omitting types for variants of first order logic can be proved algebraically by using appropriate variations on the NET [9], [21]. Indeed, one theme of this paper has been to highlight this connection:…”

Omitting types for finite variable fragments and complete representations of algebras

“…Neat reducts is a venerable old notion in cylindric algebras that is gaining some momentum lately, see e.g. [37], [42], [1], [12], [45], [24], [25], [56], [47], [51], [57], [13].…”

“…We say that T is realized in Jl if f L e r V"* ^ ^-Let y> be a formula and T be a theory. We say that <p ensures T in T if 7 1 |= <p -> y for all y € I\ The classical Henkin-Orey Omitting Types Theorem, OTT for short, [11, Theorem 2.2.9], or rather the contrapositive thereof, states that if T is a complete, consistent theory in a countable language Jz? and T{x\,..., x") C JS?…”