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We give a simple new construction of representable relation algebras with non-representable completions. Using variations on our construction, we show that the elementary closure of the class of completely representable relation algebras is not finitely axiomatizable (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

Built on the foundations laid by Peirce, Schröder, and others in the 19th century, the modern development of relation algebras started with the work of Tarski and his colleagues [21, 22]. They showed that relation algebras can capture strong first-order theories like ZFC, and so their equational theory is undecidable. The less expressive class WA of weakly associative relation algebras was introduced by Maddux [7]. Németi [16] showed that WA's have a decidable universal theory. There has been extensive research on increasing the expressive power of WA by adding new operations [1, 4, 11, 13, 20]. Extensions of this kind usually also have decidable universal theories. Here we give an example – extending WA's with set-theoretic projection elements – where this is not the case. These “logical” connectives are set-theoretic counterparts of the axiomatic quasi-projections that have been investigated in the representation theory of relation algebras [22, 6, 19]. We prove that the quasi-equational theory of the extended class PWA is not recursively enumerable. By adding the difference operator D one can turn WA and PWA to discriminator classes where each universal formula is equivalent to some equation. Hence our result implies that the projections turn the decidable equational theory of “WA + D ” to non-recursively enumerable (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

In [10] it is claimed that the set of predicate tautologies of all complete BL-chains and the set of all standard tautologies (i. e., the set of predicate formulas valid in all standard BL-algebras) coincide. As noticed in [11], this claim is wrong. In this paper we show that a complete BL-chain B satisfies all standard BL-tautologies iff for any transfinite sequence (ai: i ∈ I) of elements of B, the condition ∧i ∈ I (a2i) = (∧i ∈ Iai)2 holds in B. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

Letn > 2. A weakly representable relation algebra that is not strongly representable is constructed. It is proved that the set of all n by n basic matrices forms a cylindric basis that is also a weakly but not a strongly representable atom structure. This gives an example of a binary generated atomic representable cylindric algebra with no complete representation. An application to first order logic is given. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

Let 1 < n < ω and β > n. We show that the class NrnCAβ of n-dimensional neat reducts of β-dimensional cylindric algebras is not closed under forming elementary subalgebras. This solves a long-standing open problem of Tarski and his co-authors Andréka, Henkin, Monk and Németi. The proof uses genuine model-theoretic arguments. [ABSTRACT FROM AUTHOR]

We propose a definition of weak o-minimality for structures expanding a Boolean algebra. We study this notion, in particular we show that there exist weakly o-minimal non o-minimal examples in this setting. [ABSTRACT FROM AUTHOR]

The article presents a supplementation to the article "Finitary Algebraic Logic" that was written by the author in the previous issue of the journal. It offers corrections to errors committed in the said issue that were relevant to semiassociative relation algebra and algebraic logic. It explains the difference between a strong point-density and point-density, noting that strong point-density is preserved under homomorphisms while point-density is not.

The article presents the notion of syllogistic of a theory in a situations system. It shows the connections between the syllogistic and the relation algebras and the mathematical formulation that defines the first-order theory containing the binary predicate symbols, the syllogistic theses of a theory, and the propositions of a theory. It also cites the mathematical theorem of the syllogistics of the transitive relation theory, Keynes syllogistic, and syllogistic geometry.