Monops, Monoids and Operads: The Combinatorics of Sheffer Polynomials

We introduce a new algebraic construction, monop, that combines monoids (with respect to the product of species), and operads (monoids with respect to the substitution of species) in the same algebraic structure. By the use of properties of cancellative set-monops we construct a family of partially ordered sets whose prototypical examples are the Dowling lattices. They generalize the enriched partition posets associated to a cancellative operad, and the subset posets associated to a cancellative monoid. Their Whitney numbers of the first and second kind are the connecting coefficients of two umbral inverse Sheffer sequences with the family of powers $\{x^n\}_{n=0}^{\infty}$. Equivalently, the entries of a Riordan matrix and its inverse. This aticle is the first part of a program in progress to develop a theory of Koszul duality for monops.

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On Lattice Embeddings for Partially Ordered Sets

Canadian Journal of Mathematics ◽

10.4153/cjm-1954-057-5 ◽

1954 ◽

Vol 6 ◽

pp. 525-528

Author(s):

Truman Botts

Keyword(s):

Binary Relation ◽

Partially Ordered Sets ◽

Ordered Sets ◽

Partially Ordered ◽

The Family ◽

Transitive Binary Relation

Let P be a set partially ordered by a (reflexive, antisymmetric, and transitive) binary relation ≺. Let be the family of all subsets K of P having the property that x ∈ P and y ∈ K and y ≺ x imply x ∈ K.

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Constant Rate Distributions on Partially Ordered Sets

Journal of Probability and Statistics ◽

10.1155/2010/675754 ◽

2010 ◽

Vol 2010 ◽

pp. 1-21

Author(s):

Kyle Siegrist

Keyword(s):

Failure Rate ◽

Algebraic Structure ◽

Point Processes ◽

Probability Distributions ◽

Partially Ordered Sets ◽

Rooted Trees ◽

Ordered Sets ◽

Partially Ordered ◽

Constant Rate

We consider probability distributions with constant rate on partially ordered sets, generalizing distributions in the usual reliability setting that have constant failure rate. In spite of the minimal algebraic structure, there is a surprisingly rich theory, including moment results and results concerning ladder variables and point processes. We concentrate mostly on discrete posets, particularly posets whose graphs are rooted trees. We pose some questions on the existence of constant rate distributions for general discrete posets.

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A partial Baer *-semigroup of relations

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700029463 ◽

1975 ◽

Vol 19 (2) ◽

pp. 160-172

Author(s):

R. H. Schelp

Keyword(s):

Partially Ordered Sets ◽

Boolean Lattice ◽

Orthomodular Poset ◽

Ordered Sets ◽

Orthomodular Lattices ◽

Partially Ordered ◽

Orthomodular Posets ◽

The Family ◽

Boolean Lattices ◽

Atomic Lattices

It is shown in Gudder and Schelp (1970) that partial Baer *-semigroups coordinatize orthomodular partially ordered sets (orthomodular posets). This means for P an orthomodular poset there exists a partial Baer *-semigroup whose closed projections are order isomorphic to P preserving ortho-complementation. This coordinatization theorem generalizes Foulis (1960) in which orthomodular lattices are coordinatized by Baer *-semigroups. In particular Foulis (unpublished) shows that any complete atomic Boolean lattice is coordinatized by a Bear *-semigroup of relations. Since Greechie (1968), (1971) shows that a whole class of orthomodular posets can be formed by “pasting” together Boolean lattices, it is natural to consider the following problem. Let y be a family of Baer *-semigroups of relations which coordinatize the family B of complete atomic lattices. Is it possible to construct a partial *-semigroup of relations R which contains each member of Y such that when P is an orthomodular poset obtained by a “Greechie pasting” of members of 38 then 91 coordinatizes R This question is considered in the sequel and answered affirmatively for a certain subclass of “Greechie pasted” orthomodular posets. In addition the construction of 8)t nicely fulfills another objective in that it provides us with “nontrivial” coordinate partial Baer *-semigroups for a whole family of well known orthomodular posets. This is particularly significant since the only other known coordinate partial Baer *-semigroups, for those posets in this family which are not lattices, are the “minimal” ones given in Gudderand and Schelp (1970).

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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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