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Clifford monoids and Boolean-valued groups with zero. (Russian) Zbl 0743.20062
The aim is the construction of a non-additive analog of the known representations of a strongly regular ring in the form of a factor ring of a Boolean-valued division ring [E. Ellentuck, Fundam. Math. 96, 67-86 (1977; Zbl 0365.02044); K. Smith, J. Symb. Logic 49, 281-297 (1984; Zbl 0589.03032)]. In doing so, the Clifford monoids with zero turn out to be a natural analog of strongly regular rings and Boolean-valued groups with zero [R. Mansfield, Ann. Math. Logic 2, 297-323 (1971; Zbl 0216.29401)] are an analog of Boolean-valued division rings, the corresponding Boolean algebra \(\mathcal L\) containing the lattice \(\mathcal E\) of idempotents of the considered Clifford monoid.
MSC:
20M10 General structure theory for semigroups
06E20 Ring-theoretic properties of Boolean algebras
03C90 Nonclassical models (Boolean-valued, sheaf, etc.)
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16K40 Infinite-dimensional and general division rings
03C60 Model-theoretic algebra
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