1,354,825 research outputs found
Are interactions between different time-scales a characteristic of complexity?
No full textA self-organized complex natural system, such as a biological, a neural or a social system, is characterized by the fact that its dynamics is generated by a network of competitive regulations, each one acting as a 'simple system' (in the Newtonian sense) at a given level of complexity and with its own time-scale. A dialectics dependent on specific structural temporal constraints is established between them, punctuated by local fractures imposing a change of strategy. Such systems are capable of anticipation and adaptation thanks to the development of a memory.
The Memory Evolutive Systems (MES) defined by Ehresmann and Vanbremeersch in a series of papers since 1986 represent a mathematical model for such systems, based on the Theory of categories. This model takes into account the above properties, and it allows to study the problem of emergence; an analysis of causality attributions shows that MES satisfy the definition given by Rosen for an 'organism'
Ehresmann Semigroups from a Range Restriction Viewpoint
No full textThe first theorem in this article provides the connection between Ehresmann semigroups and range prerestriction semigroups defined by the author. By this connection, we can redefine any Ehresmann semigroups by two unary operations and eight axioms. This connection leads us to a generalization of Ehresmann’s theorem for a range prerestriction categories; as special cases, we obtain Ehresmann’s theorems for range restriction categories and for inverse categories
ON PSEUDO-EHRESMANN SEMIGROUPS
No full textAs generalizations of inverse semigroups, Ehresmann semigroups are introduced by Lawson and investigated by many authors extensively in the literature. In particular, Lawson has proved that the category of Ehresmann semigroups and admissible morphisms is isomorphic to the category of Ehresmann categories and strongly ordered functors, which generalizes the well-known Ehresmann–Schein–Nambooripad (ESN) theorem for inverse semigroups. From a varietal perspective, Ehresmann semigroups are derived from reducts of inverse semigroups. In this paper, inspired by the approach of Jones [‘A common framework for restriction semigroups and regular -semigroups’, J. Pure Appl. Algebra216 (2012), 618–632], Ehresmann semigroups are extended from a varietal perspective to pseudo-Ehresmann semigroups derived instead from reducts of regular semigroups with a multiplicative inverse transversal. Furthermore, motivated by the method used by Gould and Wang [‘Beyond orthodox semigroups’, J. Algebra368 (2012), 209–230], we introduce the notion of inductive pseudocategories over admissible quadruples by which pseudo-Ehresmann semigroups are described. More precisely, we show that the category of pseudo-Ehresmann semigroups and (2,1,1,1)-morphisms is isomorphic to the category of inductive pseudocategories over admissible quadruples and pseudofunctors. Our work not only generalizes the result of Lawson for Ehresmann semigroups but also produces a new approach to characterize regular semigroups with a multiplicative inverse transversal.</jats:p
On generalized Ehresmann semigroups
No full textAbstract
As a generalization of the class of inverse semigroups, the class of Ehresmann semigroups is introduced by Lawson and investigated by many authors extensively in the literature. In particular, Gomes and Gould construct a fundamental Ehresmann semigroup CE from a semilattice E which plays for Ehresmann semigroups the role that TE plays for inverse semigroups, where TE is the Munn semigroup of a semilattice E. From a varietal perspective, Ehresmann semigroups are derived from reduction of inverse semigroups. In this paper, from varietal perspective Ehresmann semigroups are extended to generalized Ehresmann semigroups derived instead from normal orthodox semigroups (i.e. regular semigroups whose idempotents form normal bands) with an inverse transversal. We present here a semigroup C(I,Λ,E∘) from an admissible triple (I, Λ, E∘) that plays for generalized Ehresmann semigroups the role that CE from a semilattice E plays for Ehresmann semigroups. More precisely, we show that a semigroup is a fundamental generalized Ehresmann semigroup whose admissible triple is isomorphic to (I, Λ, E∘) if and only if it is (2,1,1,1)-isomorphic to a quasi-full (2,1,1,1)-subalgebra of C(I,Λ,E∘). Our results generalize and enrich some results of Fountain, Gomes and Gould on weakly E-hedges semigroups and Ehresmann semigroups.</jats:p
On generalized Ehresmann semigroups
No full textAs a generalization of the class of inverse semigroups, the class of Ehresmann semigroups is introduced by Lawson and investigated by many authors extensively in the literature. In particular, Gomes and Gould construct a fundamental Ehresmann semigroup CE from a semilattice E which plays for Ehresmann semigroups the role that TE plays for inverse semigroups, where TE is the Munn semigroup of a semilattice E. From a varietal perspective, Ehresmann semigroups are derived from reduction of inverse semigroups. In this paper, from varietal perspective Ehresmann semigroups are extended to generalized Ehresmann semigroups derived instead from normal orthodox semigroups (i.e. regular semigroups whose idempotents form normal bands) with an inverse transversal. We present here a semigroup C(I,Λ,E∘) from an admissible triple (I, Λ, E∘) that plays for generalized Ehresmann semigroups the role that CE from a semilattice E plays for Ehresmann semigroups. More precisely, we show that a semigroup is a fundamental generalized Ehresmann semigroup whose admissible triple is isomorphic to (I, Λ, E∘) if and only if it is (2,1,1,1)-isomorphic to a quasi-full (2,1,1,1)-subalgebra of C(I,Λ,E∘). Our results generalize and enrich some results of Fountain, Gomes and Gould on weakly E-hedges semigroups and Ehresmann semigroups
On Ehresmann semigroups
No full textWe formulate an alternative approach to describing Ehresmann semigroups by means of left and right étale actions of a meet semilattice on a category. We also characterize the Ehresmann semigroups that arise as the set of all subsets of a finite category. As applications, we prove that every restriction semigroup can be nicely embedded into a restriction semigroup constructed from a category, and we describe when a restriction semigroup can be nicely embedded into an inverse semigroup.</p
Proper Ehresmann semigroups
No full textWe propose a notion of a proper Ehresmann semigroup based on a three-coordinate description of its generating elements governed by certain labelled directed graphs with additional structure. The generating elements are determined by their domain projection, range projection and -class, where denotes the minimum congruence that identifies all projections. We prove a structure result on proper Ehresmann semigroups and show that every Ehresmann semigroup has a proper cover. Our covering monoid turns out to be isomorphic to that from the work by Branco, Gomes and Gould and provides a new view of the latter. Proper Ehresmann semigroups all of whose elements admit a three-coordinate description are characterized in terms of partial multiactions of monoids on semilattices. As a consequence we recover the two-coordinate structure result on proper restriction semigroups.29 pages, revised versio
Beyond Regular Semigroups
No full textThe topic of this thesis is the class of weakly U-abundant semigroups. This class is very wide, containing inverse, orthodox, regular, ample, adequate, quasi-adequate, concordant, abundant, restriction, Ehresmann and weakly
abundant semigroups. A semigroup with subset of idempotents U is weakly U-abundant if every \art_U-class and every \elt_U-class contains an idempotent of U, where \art_U and \elt_U are relations extending the well known Green's relations \ar and \el. We assume throughout that our semigroups satisfy a condition known as the Congruence Condition (C).
We take several approaches to weakly -abundant semigroups. Our first results describe those that are analogous to completely simple semigroups. Together with an existing result of Ren this determines the structure of those weakly -abundant semigroups that are analogues of completely regular semigroups, that is, they are superabundant. Our description is in terms of a semilattice of rectangular bands of monoids.
The second strand is to aim for an extension of the Hall-Yamada theorem for orthodox semigroups as spined products of inverse semigroups and fundamental orthodox semigroups. To this end we consider weakly -orthodox semigroups, where is a band. We note that if
is a semilattice then a weakly -orthodox semigroup is exactly an Ehresmann semigroup. We provide a description of a weakly -orthodox semigroup as a spined product of a fundamental weakly -orthodox semigroup (depending only on ) and , where is isomorphic to and is the analogue of the least inverse congruence on an orthodox semigroup. This result is an analogue of the Hall-Yamada theorem for orthodox semigroups. In the case that is a normal band, or is weakly -superabundant, we find a closed form for , which simplifies our result to a straightforward form.
For the above to work smoothly in the case is weakly -superabundant, we need to find a canonical fundamental weakly -superabundant subsemigroup of . This we do, and give the corresponding answers in the case of the Hall semigroup and a number of intervening semigroups.
We then change our direction. A celebrated result of Nambooripad shows that regular semigroups are determined by ordered groupoids built over a regular biordered set. Our aim, achieved at the end of the thesis, is to extend Nambooripad's work to {\em weakly -regular}
semigroups, that is, weakly -abundant semigroups with (C) and generating a regular subsemigroup
whose set of idempotents is .
As an intervening step we consider weakly -orthodox semigroups in this light. We take two approaches. The first is via a new construction of an inductive generalised category over a band. In doing so we produce a new approach to characterising orthodox semigroups, by using inductive generalised groupoids.
We show that the category of weakly -orthodox semigroups is isomorphic to the category of inductive generalised categories over bands. Our approach is influenced by that of Nambooripad, however, there are significant differences in strategy, the first
being the introduction of generalised categories and the second being that it is more convenient to consider (generalised) categories equipped with pre-orders, rather than with partial orders. Our work may be regarded as extending a result of Lawson for Ehresmann semigroups. We also examine the trace of a weakly -orthodox semigroup, which is a primitive weakly -orthodox semigroup.
We then take a more `traditional' approach to weakly -orthodox semigroups via band categories and weakly orthodox categories over a band, equipped with two pre-orders. We show that the category of weakly -orthodox semigroups is equivalent to the category of weakly orthodox categories over bands. To do so we must substantially adjust Armstrong's method for concordant semigroups.
Finally, we consider the most general case of weakly -regular semigroups. Following Nambooripad's theorem, which establishes a correspondence between algebraic structures (inverse semigroups) and ordered structures (inductive group-oids), we build a correspondence between the category of weakly -regular semigroups and the category of weakly regular categories over regular biordered sets, equipped with two pre-orders
ON LOCALLY EHRESMANN SEMIGROUPS
No full text It was first proved by McAlister in 1983 that every locally inverse semigroup is a locally isomorphic image of a regular Rees matrix semigroup over an inverse semigroup and Lawson in 2000 further generalized this result to some special locally adequate semigroups. In this paper, we show how these results can be extended to a class of locally Ehresmann semigroups. </jats:p
Classical differential geometry with Christoffel symbols of Ehresmann -connections
No full textsummary:We give a method based on an idea of O. Veblen which gives explicit formulas for the covariant derivatives of natural objects in terms of the Christoffel symbols of a symmetric Ehresmann -connection
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