Regular semigroups of endomorphisms of ordered sets

Cross-connection is a construction of regular semigroups using certain categories called normal categories which are abstractions of the partially ordered sets of principal left (right) ideals of a semigroup. We describe the cross-connections in the semigroup [Formula: see text] of all non-invertible transformations on a set [Formula: see text]. The categories involved are characterized as the powerset category [Formula: see text] and the category of partitions [Formula: see text]. We describe these categories and show how a permutation on [Formula: see text] gives rise to a cross-connection. Further, we prove that every cross-connection between them is induced by a permutation and construct the regular semigroups that arise from the cross-connections. We show that each of the cross-connection semigroups arising this way is isomorphic to [Formula: see text]. We also describe the right reductive subsemigroups of [Formula: see text] with the category of principal left ideals isomorphic to [Formula: see text]. This study sheds light into the more general theory of cross-connections and also provides an alternate way of studying the structure of [Formula: see text].

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To the spectral theory of partially ordered sets

Sibirskii matematicheskii zhurnal ◽

10.33048/smzh.2019.60.308 ◽

2018 ◽

Vol 60 (3) ◽

pp. 578-598

Author(s):

Yu. L. Ershov ◽

M. V. Schwidefsky

Keyword(s):

Spectral Theory ◽

Partially Ordered Sets ◽

Ordered Sets ◽

Partially Ordered

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Modelling Nondeterministic Concurrent Processes with Event Structures

Fundamenta Informaticae ◽

10.3233/fi-1991-14103 ◽

1991 ◽

Vol 14 (1) ◽

pp. 39-73

Author(s):

Rita Loogen ◽

Ursula Goltz

Keyword(s):

Dynamic Behaviour ◽

Partially Ordered Sets ◽

Parallel Composition ◽

Ordered Sets ◽

Communicating Sequential Processes ◽

Partially Ordered ◽

Event Structures ◽

Transition Relation ◽

Concurrent Processes ◽

Sequential Processes

We present a non-interleaving model for non deterministic concurrent processes that is based on labelled event structures. We define operators on labelled event structures like parallel composition, nondeterministic combination, choice, prefixing and hiding. These operators correspond to the operations of the “Theory of Communicating Sequential Processes” (TCSP). Infinite processes are defined using the metric approach. The dynamic behaviour of event structures is defined by a transition relation which describes the execution of partially ordered sets of actions, abstracting from internal events.

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Estimating the area under a receiver operating characteristic curve using partially ordered sets

The International Journal of Biostatistics ◽

10.1515/ijb-2019-0127 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Ehsan Zamanzade ◽

Xinlei Wang

Keyword(s):

Roc Curve ◽

Characteristic Curve ◽

Partially Ordered Sets ◽

Sampling Technique ◽

Real Data ◽

Cost Effective ◽

Ordered Sets ◽

Operating Characteristics ◽

Partially Ordered ◽

Receiver Operating

AbstractRanked set sampling (RSS), known as a cost-effective sampling technique, requires that the ranker gives a complete ranking of the units in each set. Frey (2012) proposed a modification of RSS based on partially ordered sets, referred to as RSS-t in this paper, to allow the ranker to declare ties as much as he/she wishes. We consider the problem of estimating the area under a receiver operating characteristics (ROC) curve using RSS-t samples. The area under the ROC curve (AUC) is commonly used as a measure for the effectiveness of diagnostic markers. We develop six nonparametric estimators of the AUC with/without utilizing tie information based on different approaches. We then compare the estimators using a Monte Carlo simulation and an empirical study with real data from the National Health and Nutrition Examination Survey. The results show that utilizing tie information increases the efficiency of estimating the AUC. Suggestions about when to choose which estimator are also made available to practitioners.

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Completions of Pseudo Ordered Sets

Order ◽

10.1007/s11083-021-09565-4 ◽

2021 ◽

Author(s):

Maria D. Cruz-Quinones ◽

John Harding

Keyword(s):

Ordered Sets

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On the lattice of varieties of completely regular semigroups

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700025726 ◽

1983 ◽

Vol 35 (2) ◽

pp. 227-235 ◽

Cited By ~ 12

Author(s):

P. R. Jones

Keyword(s):

Lattice Of Varieties ◽

Regular Semigroups ◽

Completely Regular ◽

Variety Of Groups ◽

Completely Regular Semigroups ◽

Subdirect Products

AbstractSeveral morphisms of this lattice V(CR) are found, leading to decompostions of it, and various sublattices, into subdirect products of interval sublattices. For example the map V → V ∪ G (where G is the variety of groups) is shown to be a retraction of V(CR); from modularity of the lattice V(BG) of varieties of bands of groups it follows that the map V → (V ∪ V V G) is an isomorphism of V(BG).

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