Showing papers on "Cancellative semigroup published in 2019"
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TL;DR: D'Aquino and Macintyre as discussed by the authors gave axioms for a class of ordered structures, called truncated ordered abelian groups (TOAGs) carrying an addition.
Abstract: We give axioms for a class of ordered structures, called truncated ordered abelian groups (TOAG’s) carrying an addition. TOAG’s come naturally from ordered abelian groups with a 0 and a $$+$$
, but the addition of a TOAG is not necessarily even a cancellative semigroup. The main examples are initial segments $$[0, \tau ]$$
of an ordered abelian group, with a truncation of the addition. We prove that any model of these axioms (i.e. a truncated ordered abelian group) is an initial segment of an ordered abelian group. We define Presburger TOAG’s, and give a criterion for a TOAG to be a Presburger TOAG, and for two Presburger TOAG’s to be elementarily equivalent, proving analogues of classical results on Presburger arithmetic. Their main interest for us comes from the model theory of certain local rings which are quotients of valuation rings valued in a truncation [0, a] of the ordered group $${\mathbb {Z}}$$
or more general ordered abelian groups, via a study of these truncations without reference to the ambient ordered abelian group. The results are used essentially in a forthcoming paper (D’Aquino and Macintyre, The model theory of residue rings of models of Peano Arithmetic: The prime power case, 2021, arXiv:2102.00295
) in the solution of a problem of Zilber about the logical complexity of quotient rings, by principal ideals, of nonstandard models of Peano arithmetic.
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TL;DR: The main objective of as mentioned in this paper is to show that if enough small subsegments of a closed set can be found, then the product of any two closed sets is again closed.
Abstract: This paper is concerned with closure systems defined on groups such that the product of any two closed sets is again closed. The main objective of this paper is to show that if enough “small” subse...
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TL;DR: In this paper, it was shown that the monoid of all super-tropical matrices extending tropical matrices satisfies nontrivial semigroup identities and carried over to walks on labeled-weighted digraphs with double arcs.
Abstract: We prove that, for any $n$, the monoid of all $n \times n$ supertropical matrices extending tropical matrices satisfies nontrivial semigroup identities. These identities are carried over to walks on labeled-weighted digraphs with double arcs.