The Class of Neat Reducts Is Not Boolean Closed

Call a class of algebras K Boolean closed if whenever A ∈ K and B ∼= A in some Boolean valued extension of the universe of sets, then B ∈ K. SC, CA, QA and QEA stand for the classes of Pinter’s substitution algebras Tarski’s cylindric algebras, Halmos’ quasi-polyadic and quasi-polyadic equality algebras, respectively. Let 1 < n ≤ ω, n < m and K ∈ {SC,CA,QA,QEA}. We show that the class NrnKm of all neat n-reducts of algebra in Km (m-dimensional algebras in K) is not Boolean closed. We also show that NrnCAm snd NrnSCm regarded as concrete categories are not finitely complete (closed under finite limits). We mostly follow the notation and terminology of [11] which is in conformity with the monograph [3]. In particular, CAm denotes the class of cylindric algebras of dimension m. For set theory we follow the standard notation adopted in [5]. In [11] it is proved that for ordinals 1 < n < m, the class of neat n-reducts of algebras in CAm, or NrnCAm for short, is not closed under elementary subalgebras, settling a long standing open problem in algebraic logic [4][4.4]. This result is generalized for neat reducts of SC, QA and QEA in [20] and [22]. The notion of neat reducts [3][2.6.28] is in old venerable notion in cylindric algebras that was invented by Leon Henkin. This notion, together with the related notion of neat embeddings, have been revived lately and these notions occur one way or another in the following articles, (besides [11]), [7], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [6], and [23]. Neat reducts proved immensely fruitful not only for representation theory but also for such seemingly remote areas as positive solutions to the socalled finitization problem in algebraic logic 52 Tarek Sayed Ahmed [16], [17]. [19] surveys such results highlighting the important notion of neat reducts in algebraic logic. Throughout K ∈ {SC,QA,CA,QEA}. In our treatment of K we follow [23] and we follow the notation adopted therein (K is studied also in [13] and [18]. The notation used in these references is uniform.) In particular, Ksn(WKsn) denotes the class of (weak) set algebras of dimension n. In this note, we adress yet a new closure property of the class NrnKm of all neat n reducts of algebras in Km. Let M be the universe of sets and let C ∈ M be a complete Boolean algebra. (Note that C is a Boolean algebra “from the outside as well” but not necessarily complete.) Form the Boolean valued extension M of M and let ||φ|| be the Boolean value of a sentence φ of set theory containing parameters from M. φ is valid in M if ||φ|| = 1 in symbols M |= φ. Write C : A ∼= B if M |= Ā ∼= B̄. Here s̆ is the canonical name of s in M. We say that A and B are Boolean isomorphic if there is such C, and (in which case) we write A ∼=b B. It turns out that Boolean isomorphism lies somewhere between ≡ (elementary equivalence) and ∼= (isomorphism). Such an equivalence relation, as it turns out, is purely structural and can characterized by games (see below.) (The idea is to look at isomorphisms between a finite number of elements at a time.) Of course if A ∼= B then trivially A ∼=b B. And in the other direction we have: Theorem 1. If A ∼=b B, then A and B are elementary equivalent. Proof. Let L be the first order language of A and B. In [2] it is proved that A ≡∞ω B, that is L∞ω obtained from L by adding arbitrary conjunctions, does not distinguish between A and B. We now turn to defining our concrete algebras. Unless otherwise specified n is a finite cardinal > 1 and (R,+) denotes a fixed uncountable abelian group. (Any cardinal > ω will do.) Our next Theorem is almost the same as Lemma 1 in [11] except for condition (vi) of composition of n-ary relations. Theorem 2 stipulates the existence of a certain saturated first order structure which we shall use to build our algebras. (Saturation will be used to eliminate quantifiers. This in turn, renders neat reducts.) First we fix some notation: The Class of Neat Reducts is Not Boolean Closed 53