Logics that are generated by idempotents. (English) Zbl 1060.15018

The idempotence properties of the rings of square matrices are studied. A series of special definitions is given (a logic or an orthomodular coset, orthomodular lattice, a subset of logic, a Boolean subalgebra, a generalized \(R\)-Latin square, a \(R\)-circulant). The generalized theorem which represents one of the cases concerning the characteristics of the logic of idempotents is proved [cf. V. V. Kalinin, Algebra Logika 15, 535–557 (1976; Zbl 0377.06003); F. Katrnoška, Logics and states of physical systems (in Czech). Thesis, Czech Technical University, Prague (1980); J. Flachsmeyer, Topology and measure III, Proc. Conf, Vitte/Hiddensee 1980, Part 1, 65–73 (1982; Zbl 0536.06002)]. \(R\)-circulant matrices in the framework of the ring \(R\) are introduced and some consequences for the logic of idempotents of the corresponding rings are considered. Some examples are indicated. Particularly, the connection between rings of \(2\times 2\)-matrices and spin matrices is briefly discussed.


15B33 Matrices over special rings (quaternions, finite fields, etc.)
16S50 Endomorphism rings; matrix rings
06C15 Complemented lattices, orthocomplemented lattices and posets
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