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The case of Piaget’s group INRC. (English) Zbl 0569.92029

Author’s summary: What is it that distinguishes J. Piaget’s [Arch. Sci., Genève 2, 179-182 (1949; Zbl 0034.007) and ibid. 3, 159-161 (1950; Zbl 0037.294)] transformations N, R, and C from the rest of the 16! transformations of the 16 binary propositional operations? Here Piaget’s INRC is considered as a subgroup of the group \({\mathcal M}_ 2\) of all automorphisms and dual automorphisms of the free Boolean algebra with two generators. This group is isomorphic to \(S_ 4\times C_ 2\). Its elements are given explicitly.
Many other psychologically relevant subgroups of \({\mathcal M}_ 2\) play an important role. They are discussed and their connections shown. Particular attention is given to involutions, even if the view that they constitute the sole representation of reversibility is abandoned.
Piaget’s transformation R turns out not to be the inverse operation of relations. The group of automorphisms, dual automorphisms, anti- automorphisms of the algebra of binary relations on a finite set is found.
A crystallographic presentation of these groups is given and related work by W. M. Bart, J. Math. Psychol. 8, 539-553 (1971; Zbl 0228.92008), Leresche, Rev. Europ. Sci. Soc. 14, 219-241 (1976), and G. Pólya, J. Symb. Logic, 5, 98-103 (1940; Zbl 0024.00102) is discussed.
Reviewer: M.Eytan

MSC:

91E99 Mathematical psychology
03G05 Logical aspects of Boolean algebras
20B99 Permutation groups
20C30 Representations of finite symmetric groups
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