Pregroups and natural language processing (Q1002122)

The author proposes residuated monoids (partially ordered monoids with two binary operations \(/\) and \(\backslash\)\ abiding by the rule \[ ab\rightarrow c\Longleftrightarrow a\rightarrow c/b\Longleftrightarrow b\rightarrow a\backslash c \] as a mathematical framework for grammars of natural languages [the author, Cat. Grammar, 153--172 (1988; Zbl 0692.03019)]. A residuated monoid with lattice operations becomes a Grishin algebra if a so-called dualizing element \(0\), subject to the conditions \[ \left( 0/a\right) \backslash0=a=0/\left( a\backslash0\right), \] \ is added, in which the de Morgan dual \(+\) to the jaxtaposition standing for the monoid operation is defined by \[ a+b=\left( \left( 0/b\right) \left( 0/a\right) \right) \backslash0 \] A pregroup is a Grishin algebra in which \[ 0=1\text{ and }a+b=ab \] hold. A residuated monoid is a semi-Heyting algebra if it satisfies \[ a\rightarrow1, a\rightarrow aa\text{ and }ab\rightarrow ba \] From a logical point of view, free residuated monoids, Grishin algebras, pregroups and semi-Heyting algebras correspond to syntactic calculus, classical bilinear logic, compact bilinear logic and positive intuitionistic propositional calculus respectively. This expository paper, which is an expanded version of the author's [Mathematics and its applications (Dordrecht) 565, 389--397 (2004; Zbl 1078.18004)], explains why the author has changed his belief from [Zbl 0692.03019] to pregroups.