Omitting types for finite variable fragments and complete representations of algebras

We give a novel application of algebraic logic to first order logic. A new, flexible construction is presented for representable but not completely representable atomic relation and cylindric algebras of dimension n (for finite n > 2) with the additional property that they are one-generated and the set of all n by n atomic matrices forms a cylindric basis. We use this construction to show that the classical Henkin-Orey omitting types theorem fails for the finite variable fragments of first order logic as long as the number of variables available is > 2 and we have a binary relation symbol in our language. We also prove a stronger result to the effect that there is no finite upper bound for the extra variables needed in the witness formulas. This result further emphasizes the ongoing interplay between algebraic logic and first order logic.

Relation algebra reducts of cylindric algebras and complete representations

We show, for any ordinal γ ≥ 3, that the class ℜaCAγ is pseudo-elementary and has a recursively enumerable elementary theory. ScK denotes the class of strong subalgebras of members of the class K. We devise games, Fn (3 ≤ n ≤ ω), G, H, and show, for an atomic relation algebra with countably many atoms, thatfor 3 ≤ n < ω. We use these games to show, for γ > 5 and any class K of relation algebras satisfyingthat K is not closed under subalgebras and is not elementary. For infinite γ, the inclusion ℜaCAγ ⊂ ScℜaCAγ is strict.For infinite γ and for a countable relation algebra we show that has a complete representation if and only if is atomic and ∃ has a winning strategy in F (At()) if and only if is atomic and ∈ ScℜaCAγ.

Groups and Algebras of Binary Relations

In 1941, Tarski published an abstract, finitely axiomatized version of the theory of binary relations, called the theory of relation algebras. He asked whether every model of his abstract theory could be represented as a concrete algebra of binary relations. He and Jónsson obtained some initial, positive results for special classes of abstract relation algebras. But Lyndon showed, in 1950, that in general the answer to Tarski's question is negative. Monk proved later that the answer remains negative even if one adjoins finitely many new axioms to Tarski's system. In this paper we describe a far-reaching generalization of the positive results of Jónsson and Tarski, as well as of some later, related results of Maddux. We construct a class of concrete models of Tarski's axioms—called coset relation algebras—that are very close in spirit to algebras of binary relations, but are built using systems of groups and cosets instead of elements of a base set. The models include all algebras of binary relations, and many non-representable relation algebras as well. We prove that every atomic relation algebra satisfying a certain measurability condition—a condition generalizing the conditions imposed by Jónsson and Tarski—is essentially isomorphic to a coset relation algebra. The theorem raises the possibility of providing a positive solution to Tarski's problem by using coset relation algebras instead of the standard algebras of binary relations.

Negative-existentially complete structures and definability in free extensions

Let R be a commutative ring with 1 and R[X1, …, Xn] the polynomial ring in n variables over R. Then for any relation f(X) = 0 in R[X] there exists a conjunction of equations φf such that f(X) = 0 holds in R[X] iff φf holds in R; φf is of course the formula saying that all the coefficients of f(X) vanish. Moreover, φf is independent of R and formed uniformly for all polynomials f up to a given formal degree. In this paper we investigate first order theories T for which a similar phenomenon holds. More precisely, we let TAH be the universal Horn part of a theory T and look at free extensions of models of T in the class of models of TAH. We ask whether an atomic relation t1(X, a) = t2(X, a) or R(t1(X, a), …, tn(X, a)) in can be equivalently expressed by a finite or infinitary formula φ(a) in , such that φ(y) depends only on ti{X, y) and not on or a1, …, am ∈ A.We will show that for a wide class of theories T “defining formulas” φ(y) in this sense exist and can be taken as infinite disjunctions of positive existential formulas.

On complete atomic proper relation algebras

The object of this note is to classify the isomorphism types of all complete atomic proper relation algebras. In particular we find that the number of abstractly distinct complete atomic proper finite relation algebras of 2m elements is equal to the number of distinct partitions of the positive integer m into summands which are perfect squares.A relation algebra is called complete atomic if its Boolean algebra is complete atomic. A proper relation algebra is a relation algebra whose elements are relations. Lyndon has recently proved, by constructing an appropriate finite relation algebra, that not every complete atomic relation algebra is isomorphic to a proper relation algebra.Henceforth, let us consider a complete atomic proper relation algebra. The atoms {〈a, b〉}, {〈c, d〉} will be called connected if and only if {a, b} ⋂ {c, d} is not empty. Let the relation of connectedness over the set B of all atoms be denoted by ~. For brevity let B = {b1,b2, …}. The relation ~ is obviously reflexive and symmetric. But ~ is not transitive, since a, b, c, d distinct implies {〈a, 0〉} ~ {〈b, c〉}, {〈b, c〉} ~ {〈c, d〉}, and {〈a, b〉} ≁ {〈c, d〉}.

I. On the chemical action of water on soluble salts

Before extending my researches on chemical affinity among substances in solution, it seemed desirable to ascertain, if possible, what specific chemical action water exerts on a salt. This inquiry is beset with unusual difficulties, and unfortunately my experiments have not led to any conclusive result. Yet some of the observations made during the course of the inquiry have a value independent of theory, and a brief notice of them may not perhaps be deemed unworthy of a place in the Proceedings of the Royal Society. It is well known that many anhydrous salts will absorb water, and still remain solid bodies, either amorphous or crystallized. In such a case the water combined is always in simple atomic relation with the salt itself; great heat is often evolved, and a change of colour frequently ensues. These "hydrated" salts (as they are usually considered) are generally soluble in water; and it is the condition of such a body when dissolved that opens a wide field for speculation. The water may act merely as a solvent; or it may unite without decomposition with the dissolved salt, becoming an integral part of the compound in solution; or reciprocal decomposition may ensue, each electro-positive element combining with each electro-negative one in certain proportions; or the ultimate result may be due to two or more of these modes of action in conjunction.