On Logical Analysis of Relativity Theories

The aim of this paper is to give an introduction to our axiomatic logical analysis of relativity theories.Comment: 19 pages, 1 figure

Thr variety of coset relation algebras

A coset relation algebra is one embeddable into some full coset relation algebra, the latter is an algebra constructed from a system of groups, a coordinated system of isomorphisms between quotients of these groups, and a system of cosets that are used to "shift" the operation of relative multiplication. We prove that the class of coset relation algebras is equationally axiomatizable (that is to say, it is a variety), but no finite set of equations suffices to axiomatize the class (that is to say, the class is not finitely axiomatizable).Comment: This is the fifth member of a series of papers on measurable relation algebras. Forthcoming in The Journal of Symbolic Logic. arXiv admin note: text overlap with arXiv:1804.0027

A representation theorem for measurable relation algebras with cyclic groups

A relation algebra is measurable if the identity element is a sum of atoms, and the square $x;1;x$ of each subidentity atom $x$ is a sum of non-zero functional elements. These functional elements form a group $G_x$. We prove that a measurable relation algebra in which the groups $G_x$ are all finite and cyclic is completely representable. A structural description of these algebras is also given

A representation theorem for measurable relation algebras

A relation algebra is called measurable when its identity is the sum of measurable atoms, where an atom is called measurable if its square is the sum of functional elements. In this paper we show that atomic measurable relation algebras have rather strong structural properties: they are constructed from systems of groups, coordinated systems of isomorphisms between quotients of the groups, and systems of cosets that are used to “shift” the operation of relative multiplication. An atomic and complete measurable relation algebra is completely representable if and only if there is a stronger coordination between these isomorphisms induced by a scaffold (the shifting cosets are not needed in this case). We also prove that a measurable relation algebra in which the associated groups are all finite is atomic. © 2018 Elsevier B.V

Term algebras of elementarily equivalent atom structures

We exhibit two relation algebra atom structures such that they are elementarily equivalent but their term algebras are not. This answers Problem 14.19 in Hirsch and Hodkinson’s text Relation Algebras by Games. © 2018, Springer Nature Switzerland AG