Some varieties containing relation algebras

Abstract

ABSTRACT. Three varieties of algebras are introduced which extend the variety RA of relation algebras. They are obtained from RA by weakening the associative law for relative product, and are consequently called nonassociative, weakly-associative and semiassociative relation algebras, or NA, WA, and SA, respectively. Each of these varieties arises naturally in solving various problems concerning relation algebras. We show, for example, that WA is the only one of these varieties which is closed under the formation of complex algebras of atom structures of algebras, and that WA is the closure of the variety of representable RA's under relativization. The paper also contains a study of the elementary theories of these varieties, various representation theorems, and numerous examples.O. Introduction. Relation algebras (RA 's) have a binary operation; which serves as an abstract algebraic analogue of the relative product of binary relations. (The relative product of R, S C U X U, is R I S = {(x, z):(x, y) E Rand (y, z) E S for some y E U}.) The relative product is associative, and one of the postulates for RA's is that ; is associative. The associativity of relative product can be expressed by a sentence in a first-order language with binary relation symbols, namelyAlthough this sentence has three variables, it cannot be proved from the ordinary axioms of first-order logic without using four variables. In contrast, all the other postulates for relation algebras can not only be expressed but proved using only three variables. (These facts were first proved by Tarski. For a proof that (1) requires four variables to prove, see [3].) Tarski asked whether there are any other equations whose translations into first-order sentences can be expressed and proved using only three variables, but which are not derivable from the postulates for

Strongly representable atom structures of relation algebras

“…We show that the class of all strongly representable relation algebra atom structures is not closed under ultraproducts and is therefore not elementary. This answers a question of Maddux (1982).Our proof is based on the following construction. From an arbitrary undirected, loop-free graph Γ, we construct a relation algebra atom structure α(Γ) and prove, for infinite Γ, that α(Γ) is strongly representable if and only if the chromatic number of Γ is infinite.…”

“…We show that the class of all strongly representable relation algebra atom structures is not closed under ultraproducts and is therefore not elementary. This answers a question of Maddux (1982).…”

Some Varieties Containing Relation Algebras

“…It follows that (x;y)U = yU ;x u . By (13), (8), and the IL, we have l'u = I'u; l'uu = (I'U; I')U = l'uu = I', and hence, by (10), O'U= (-l')U = _l'u= -1'= 0'. Thus (14) holds.…”

“…For (8) We get x Uu = x by first replacingy by -x, and then by _X UU . Note that 1 = 1 + I U , so by (6) and (8) we get (9) as follows: 1 = 1 + I U = I UU + I U = (IU + I)u = I U • For (10) we first get -x . x Uu = 0 from -x' x = 0 by (8), and then (-x)U 'x u = 0 by 1.11.…”

“…(11) follows from (6) and (10), and (12) follows from (6) and (8 Let z = _(x;y)U , and then let z = _(yU ;X U ). It follows that (x;y)U = yU ;x u .…”