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1. Background

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Abstract

Generally speaking, this book deals with arithmetical systems that arise naturally as invariants of various mathematical structures.

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Fußnoten
1
A notable exception to commutativity occurs with ordinal algebras (cf. Tarski [110]).
 
2
Throughout this work, “countable” will always mean “at most countable”.
 
3
We will always write (syntactical) objects in sans serif fonts, such as x, y, z, p, …while keeping the notation x, y, z, p, …for the objects that they interpret, denizens of a given mathematical structure.
 
4
A ring R is regular if its multiplicative semigroup is regular, that is, for all xR there exists yR such that x = xyx.
 
5
Exel [41] also uses the term “Boolean inverse semigroup”, but in a weaker sense than the one used here.
 
6
Recall that a lattice (L,∨,∧) is modular if x ∧ ( yz) = (xy) ∨ z whenever x, y, zL with zx.
 
7
A lattice (L, ∨, ∧) is distributive if x ∧ ( yz) = (xy) ∨ (xz) for all x, y, zL.
 
8
A topological space is zero-dimensional if it has a basis of clopen (i.e., simultaneously closed and open) sets.
 
9
This definition is tailored to accommodate both cases of simple commutative monoids and simple partially ordered Abelian groups.
 
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Metadaten
Titel
Background
verfasst von
Friedrich Wehrung
Copyright-Jahr
2017
DOI
https://doi.org/10.1007/978-3-319-61599-8_1