The objective of these two lecturesis to present:
(1) Characterizations of the Real Semigroups (RSs) whose space of characters is Boolean in its natural spectral topology, originally introduced in the dual category of abstract real spectra in sections 7.6 and 8.9 of [M] , therein called zero-dimensional real spectra and here referred to as Boolean RSs;
(2) To give a natural Horn-geometric axiomatization of Boolean RSs and establish the closure of this class by certain important constructions: Boolean powers, arbitrary filtered colimits, arbitrary reduced products, RS-sums and by surjective RS-morphisms;
(3) To characterize the unitary commutative rings whose associated RS is Boolean;
(4) Give a natural representation of morphisms between Boolean RSs;
(5) To characterize quotients in the class of Boolean RSs.
In the first lecture, Boolean Real Semigroups, I, we shall deal with (2), (3) and parts of (1).
In the second lecture, Boolean Real Semigroups II, we shall complete the discussion of (1) (characterization of Boolean RSs), as well as treat questions (4) and (5).
The results presented in these lectures where published in [MR].
References:
[M] M. Marshall, "Spaces of Orderings
and Abstract Real Spectra", Lecture Notes in Mathematics 1636,
Springer-Verlag, Berlin, 1996.
[MR] F. Miraglia, H. Ribeiro, "Boolean Real Semigroups", Tribute Series, 44, London College Publications, 136 -175, 2021.
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