The domino relation: Flattening a two-dimensional logic (Q1120562)

The axiomatization and completeness problems for Humberstone's necessity operator [\textit{I. L. Humberstone}, Rep. Math. Logic 13, 33-42 (1981; Zbl 0519.03009)] are solved in this paper. The domino structure on a non-empty set D is defined as a pair \((D^ 2,<)\), where \((x,y)<(x',y')\) iff \(y=x'\), and the surrogate domino structure as a pair (W,R), with \(W\neq \emptyset\) and \(R\subset W^ 2\), satisfying the following conditions: \(\forall u\forall v\exists !w(uRw\wedge wRv)\), \(\forall u\exists v(uRv\wedge \forall w(uRw\supset vRw))\) and \(\forall u\exists v(uRv\wedge \forall w(vRw\supset uRw))\). The author presents an axiomatic system for the logic of absolute necessity (which can be considered as an extension of the modal logic K) and proves its completeness with respect to the class of surrogate domino structures.