Talk:⋖

RFV discussion: July–August 2015

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⋖

Defined as "(set theory) covers; is covered by", with a pointer to w:Hasse diagram. Intriguing, any citations? — Keφr 11:57, 15 July 2015 (UTC)

In discussing some other arrow-y symbols, @msh210 mentioned that he had seen this symbol but couldn't recall where. So, it's not made up. Perhaps someone with a clearer idea of how it would be used could search Google Books for strings that might be scannos of it, and/or search for works about Hasse diagrams. - -sche (discuss) 03:52, 16 July 2015 (UTC)

I've found two search engines that can handle mathematical notation if input in LaTeX: searchonmath, through which I notice that this symbol is ascribed a (different) meaning by Wikipedia's article on the Wirth–Weber precedence relationship, and latexsearch, which finds two journal articles that use this term in the following "sentence fragments":

• Tail Asymptotics for Discrete Event Systems, in Discrete Event Dynamic Systems 18 (2008 September 30), pages 563-584:
  • $[i]\lessdot [j]$ (or as quoted by latexsearch: $[i]\lessdot [j{\kern.8pt}]$)
  • ${\mathcal {C}}_{\ell }\lessdot {\mathcal {C}}_{m}$
  • $[i]\lessdot {\mathcal {C}}_{\ell }\lessdot [j]$
  • $[j]\lessdot [i]$ (or as quoted by latexsearch: $[j{\kern.8pt}]\lessdot [i]$)
  • ${\mathcal {C}}_{\ell }\lessdot {\mathcal {C}}_{m}$
• A Study on the Inequalities for Fast Similarity Search in Metric Spaces, in Trends in Communication Technologies and Engineering Science 33 (2009 January 1), pages 307-321:
  • \begin{aligned}LAESA &\lessdot AESA \lessdot LPAESA2D\\ & \lessdot \;LPAESA3D \lessdot PAESA2D \lessdot PAESA3D,\end{aligned}

- -sche (discuss) 04:12, 16 July 2015 (UTC)

More citations (of some sense):

• 2003, V E Marenich, Conjugation properties in incidence algebras, in Fundamentalnaya i Prikladnaya Matematika (Fundamental and applied mathematics), volume 9, number 3:

  Incidence algebras can be regarded as a generalization of full matrix algebras. We present some conjugation properties for incidence functions. The list of results is as follows: a criterion for a convex-diagonal function f to be conjugated to the diagonal function fe; conditions under which the conjugacy $f\sim Ce+\zeta _{\lessdot }$ holds (the function $Ce+\zeta _{\lessdot }$ may be thought of as an analog for a Jordan box from matrix theory); a proof of the conjugation of two functions $z<$ and $\zeta _{\lessdot }$ for partially ordered sets that satisfy the conditions mentioned above; an example of a partially ordered set for which the conjugacy $\zeta _{<}\sim \zeta _{\lessdot }$ does not hold. These results involve conjugation criteria for convex-diagonal functions of some partially ordered sets.

• 2014, Daniel Alpay, Maria Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa, Bicomplex and Hyperbolic Numbers, in Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis:

  Let us now define in $({\mathbb {D} })$ the following binary relation: given $(\alpha _{1}),(\alpha _{2}\in {\mathbb {D} })$, we write $(\alpha _{1}\lessdot \alpha _{2})$ whenever $(\alpha _{2}-\alpha _{1}\in {\mathbb {D} }^{+})$. It is obvious that this relation is reflexive, transitive, and […]

• 2014, T Chatain, S Haar, A canonical contraction for safe Petri nets, in Transactions on Petri Nets and Other Models of Models of Concurrency IX, in Lecture Notes in Computer Science, volume 8910, pages 83-98:

  We need a few definitions to introduce them. Denote by $(\lessdot )$ the direct causality relation defined as: for any transitions $(s)$ and $(t)$, […]

- -sche (discuss) 05:34, 27 July 2015 (UTC)

If you search Google Scholar for "lessdot", you find plenty of citations; the hard part is working out what they mean. - -sche (discuss) 05:50, 27 July 2015 (UTC)

I'm pretty sure the Marenich citation is using the challenged sense, based on the definitions earlier in the article. The Chatain and Haar citation is either using this sense or a sense that is very conceptually similar to it. The Alpay et al. citation, the 2008 citation, and the 2009 citations are using different senses.

If the challenged sense does fail, I think these five quotations are sufficient for a sense along the lines of "Used to indicate a relation conceptually or formally related to the less than relation." Different authors evidently use the symbol to mean different things, but all of the uses I've seen are somehow related to <. —Mr. Granger (talk • contribs) 12:04, 27 July 2015 (UTC)

Aha, found something which is clearly using the challenged sense:

• The order of birational motion (MIT):

  We say that u ∈ P is covered by v ∈ P (written u ⋖ v) if we have u < v and there is no w ∈ P satisfying u < w < v.

This outright says what that implies, namely that it's only "covered by", not "covers":

• Finite Posets:

  Here, ⋖ and ⋗ mean (respectively) “covered by” and “covers”, […]

This, on the other had, is something to do with logic:

• 2014, Jan van Eijck, Dynamic epistemic logics, in (Johan van Benthem on Logic and Information Dynamics, in) Outstanding Contributions to Logic, volume 5 (2014), pp 175-202:

  The S is serial, for let x be an arbitrary member of the state set A. If there is no y ∈ A with (x, y) ∈ R_α we have (x, x) ∈ S. If there is such a y then (x, y) ∈ S. So in any case there is a z ∈ A with (x, z) ∈ S. It is also easy to see that S is transitive and euclidean. Therefore (?[a]⊥;?⊤)∪ a;(aˇ;a)* can serve as a KD45 operator, and we have an appropriate way to interpret KD45 belief in epistemic PDL. Abbreviate this operator as ⋖_α.

- -sche (discuss) 16:51, 27 July 2015 (UTC)

RFV-passed. - -sche (discuss) 02:37, 6 August 2015 (UTC)