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Pro-aperiodic monoids via saturated models [article]

Samuel J. v. Gool, Benjamin Steinberg
2017 arXiv   pre-print
We apply Stone duality and model theory to study the structure theory of free pro-aperiodic monoids.  ...  Stone duality implies that elements of the free pro-aperiodic monoid may be viewed as elementary equivalence classes of pseudofinite words.  ...  In this paper, we use Stone duality and saturated models to obtain most of the known structural results about free pro-aperiodic monoids, as well as some new ones.  ... 
arXiv:1609.07736v2 fatcat:ma3tvz37ercrdon3ejdlrq5rty

Editors: Heribert Vollmer and Brigitte Vallée; Article No. 39

Samuel Gool, Benjamin Steinberg, Samuel, J Gool, Benjamin Steinberg
licensed under Creative Commons License CC-BY 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)   unpublished
We apply Stone duality and model theory to study the structure theory of free pro-aperiodic monoids.  ...  Model theory provides us with saturated words in each such class, i.e., words in which all possible factorizations are realized.  ...  Since finite words and W 3 are weakly 39:8 Pro-Aperiodic Monoids via Saturated Models saturated, the word W 3 is in fact ω-saturated. Since |W 3 | is countable, W 3 is countably saturated.  ... 
fatcat:v5sone23ofcfll3ugxc52vjz34

The linear nature of pseudowords

Jorge Almeida, Alfredo Costa, José Carlos Costa, Marc Zeitoun
2019 Publicacions matemàtiques  
We show that, in the case of the pseudovariety of aperiodic finite semigroups, the pseudoword can be recovered from the labeled linear order.  ...  They worked specially with saturated models. In our paper, the models that appear in the image of the representation are not saturated in general.  ...  Consider the pseudoword a ω of the free pro-aperiodic semigroup Ω {a} A. Then F(a ω ) has only one stationary point, namely (a ω , a ω ).  ... 
doi:10.5565/publmat6321901 fatcat:o2kcos6iu5cvxjcigvsgosyiye

The linear nature of pseudowords [article]

Jorge Almeida, Alfredo Costa, José Carlos Costa, Marc Zeitoun
2017 arXiv   pre-print
We show that, in the case of the pseudovariety of aperiodic finite semigroups, the pseudoword can be recovered from the labeled linear order.  ...  They worked specially with saturated models. In our paper, the models that appear in the image of the representation are not saturated in general.  ...  Consider the pseudoword a ω of the free pro-aperiodic semigroup Ω {a} A. Then F(a ω ) has only one stationary point, namely (a ω , a ω ).  ... 
arXiv:1702.08083v2 fatcat:ehnnxkdohbhzdeuvgvwtnp5bom

A Cook's tour of duality in logic: from quantifiers, through Vietoris, to measures [article]

Mai Gehrke, Tomas Jakl, Luca Reggio
2020 arXiv   pre-print
To be more precise, (Schützenberger, 1965) shows that the star-free languages are precisely those recognised by (finite) aperiodic monoids.  ...  However, it was introduced, not via logic and duality, but rather via the connection between automata and finite semigroups, where the multiplication available on the profinite monoid also plays a fundamental  ...  the propositional logic having as atoms p ≥q ϕ and p <q ϕ, for each ϕ ∈ FO ω and q ∈ [0, 1] ∩ Q, and the following inference rules (along with the usual ones for the Boolean connectives): The intended models  ... 
arXiv:2007.15415v1 fatcat:d7fs2d66tffg3m7o5ob6xz6tt4

Noncommutative stable homotopy and stable infinity categories

Snigdhayan Mahanta
2015 Journal of Topology and Analysis (JTA)  
Along the way we obtain infinity categorical models for some well-known bivariant homology theories like KK-theory, E-theory, and connective E-theory via suitable (co)localizations.  ...  In a subsequent project we are going to construct a model structure on Pro ω (SC * δ ), whose homotopy category will contain HoSC * as a full subcategory.  ...  There is a monoidal exact p-localization functor NSH → NSH p .  ... 
doi:10.1142/s1793525315500077 fatcat:zxlajtrcyrblrg4duzayvdn7ca

Profinite semigroups and applications [chapter]

Jorge Almeida
Structural Theory of Automata, Semigroups, and Universal Algebra  
[79] ) A language over a finite alphabet is star-free if and only if it is recognized by a finite aperiodic monoid.  ...  Hence we have N ↔ N via Eilenberg's correspondence.  ... 
doi:10.1007/1-4020-3817-8_1 fatcat:xix2tppzybhgli5gmxzqbmdkf4

Towards a pseudoequational proof theory [article]

Jorge Almeida, Ondřej Klíma
2017 pre-print
clearly sound, is also shown to be complete in many instances, such as when Σ defines a locally finite variety, a pseudovariety of groups, more generally, of completely simple semigroups, or of commutative monoids  ...  Note also that a finite monogenic additive monoid M carries a natural structure of semiring via the natural additive homomorphism N → M .  ...  Our first example is given by the pseudoidentity x ω+1 = x ω , which defines the pseudovariety A of all finite aperiodic monoids.  ... 
doi:10.4171/pm/2012 arXiv:1708.09681v1 fatcat:pzw6pvxqhjhhtfymz4x6roxh7i

Collaborating institutions [chapter]

2016 The Financial System We Need  
We define the distribution of the process through an irreducible aperiodic matrix, which we can suppose stochastic.  ...  The requirement of advanced skilled professionals has led many people to study via web.  ...  The model for noisy images is: let f be the observed image and let us assume that f =f +n, being f a noise-free image n a noise signal (n ∼ N (0, σ)).  ... 
doi:10.18356/b4968a28-en fatcat:v75hvlhqebfglfsx7q4qwwlc34

Limits to parallel computation: P-completeness theory

1996 ChoiceReviews  
The re- A.8.27 Aperiodic Monoid Membership Variety B 2 A.9 Geometry (AMonMEMB) A.9.1 Plane Sweep Triangulation (PST) Given: duction involves constructing "geometric" gadgets for routing (left and right  ...  Reference: Beaudry, McKenzie, and Thérien [29].Hint: The reduction is from a variant of Generability, Problem A.6.7.Remarks: The lattice of aperiodic monoid varieties is partitioned in [29] into the following  ... 
doi:10.5860/choice.33-3959 fatcat:qjoueeu225gr7jvwdf6gkoxfbq

Some Fundamental Theorems in Mathematics [article]

Oliver Knill
2022 arXiv   pre-print
It is for cancellative monoids (the statement a * x = b * x implies a = b in the monoid) that the monoid is also contained isomorphically inside the group.  ...  The monoid structure belongs to a collection of mathematical structures magmas ⊃ semigroups ⊃ monoids ⊃ groups. A monoid is commutative, if x * y = y * x for all x, y ∈ X.  ... 
arXiv:1807.08416v4 fatcat:lw7lbsxyznfrnaozilxapihmdy

On Graphs, Groups and Geometry [article]

Oliver Knill
2022 arXiv   pre-print
The pro-finite completion of G has fractal Hausdorff dimension 5/8.  ...  Using only a monoid structures for example and to look for the natural numbers (N, +), would not an option for us.  ... 
arXiv:2205.14097v1 fatcat:6fmpzkmuhzfrbcy4n466jw2iem