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Linear functionals on idempotent spaces: an algebraic approach [article]

Grigori Litvinov, Victor Maslov, Grigori Shpiz
<span title="2000-12-29">2000</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We consider idempotent versions of certain basic results of linear functional analysis, including the theorem on the general form of a linear functional and the Hahn-Banach and Riesz-Fischer theorems.  ...  In this paper, we present an algebraic approach to idempotent functional analysis, which is an abstract version of idempotent analysis.  ...  ACKNOWLEDGMENTS This work was supported by the INTAS and Russian Foundation for Basic Research, joint grant no. 95-91.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0012268v1">arXiv:math/0012268v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fnsxlnlbrrdtxem5tu2pr6do54">fatcat:fnsxlnlbrrdtxem5tu2pr6do54</a> </span>
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Associative and preassociative functions

Jean-Luc Marichal, Bruno Teheux
<span title="2014-02-12">2014</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ealgne43xjhvfetoaem5md3khq" style="color: black;">Semigroup Forum</a> </i> &nbsp;
We investigate the associativity property for functions of indefinite arities and introduce and discuss the more general property of preassociativity, a generalization of associativity which does not involve  ...  any composition of functions.  ...  This research is supported by the internal research project F1R-MTH-PUL-12RDO2 of the University of Luxembourg.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00233-014-9580-5">doi:10.1007/s00233-014-9580-5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/v6e5pkw6rrhmzoayox74lnnztq">fatcat:v6e5pkw6rrhmzoayox74lnnztq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170921212346/http://orbilu.uni.lu/bitstream/10993/15132/1/AssociativePreassociativeFunctions.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/fb/47/fb474687e31faee12a2c5b87ab6b3726443ef919.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00233-014-9580-5"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Page 2930 of Mathematical Reviews Vol. , Issue 80H [page]

<span title="">1980</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
For every countable set G of functions on an infinite set M there exists a set consisting of three unary idempotent functions and sixteen binary associative idempotent functions which is complete for G  ...  Jan Chvalina (Brno) Henno, Jaak 80h:08006 On the completeness of associative idempotent functions. Z. Math. Logik Grundlag. Math. 25 (1979), no. 1, 37-43.  ... 
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Preassociative aggregation functions

Jean-Luc Marichal, Bruno Teheux
<span title="">2015</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wcbukkuh2vbrpcbz2f4z2a2g6u" style="color: black;">Fuzzy sets and systems (Print)</a> </i> &nbsp;
These axiomatizations are based on existing characterizations of some noteworthy classes of associative operations, such as the class of Acz\'elian semigroups and the class of t-norms.  ...  The classical property of associativity is very often considered in aggregation function theory and fuzzy logic.  ...  Based on a recent investigation of associativity (see [7, 8] ), we show that an associative function F : X * → X is completely determined by its first two components F 1 and F 2 .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.fss.2014.11.010">doi:10.1016/j.fss.2014.11.010</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/mxwut437y5gwri3o6ndbs5zwqe">fatcat:mxwut437y5gwri3o6ndbs5zwqe</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170808141511/http://orbilu.uni.lu/bitstream/10993/17468/1/PreassociativeAggregationFunctions.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/f8/3c/f83c51511fd100f3f1a43680079ce307f182ae27.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.fss.2014.11.010"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Preassociative Aggregation Functions [chapter]

Jean-Luc Marichal, Bruno Teheux
<span title="">2014</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/jyopc6cf5ze5vipjlm4aztcffi" style="color: black;">Communications in Computer and Information Science</a> </i> &nbsp;
We investigate the associativity property for varying-arity aggregation functions and introduce the more general property of preassociativity, a natural extension of associativity.  ...  We discuss this new property and describe certain classes of preassociative functions.  ...  Based on a recent investigation of associativity (see [7, 8] ), we show that an associative function F : X * → X is completely determined by its first two components F 1 and F 2 .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-08852-5_34">doi:10.1007/978-3-319-08852-5_34</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/o6pize3mjvab5f27x4jarwmbuq">fatcat:o6pize3mjvab5f27x4jarwmbuq</a> </span>
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Incidence algebras on generalized semigroups. I

Melvin Tainiter
<span title="">1971</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/z77xaqun7bcxjkh75wb7iseaty" style="color: black;">Journal of combinatorial theory. Series A</a> </i> &nbsp;
The rank of the zeta function is studied for these structures. Conditions for closure in the incidence algebras are given.  ...  The concept of incidence algebras over partially ordered sets is generalized to include algebraic structures such as semigroups, some groupoids, and partial groupoids.  ...  If zeta is of full rank the determinant of zeta is one and the only complete matching is the identity.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0097-3165(71)90042-2">doi:10.1016/0097-3165(71)90042-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nfwtgogxmjaerhmh3onn3cplgy">fatcat:nfwtgogxmjaerhmh3onn3cplgy</a> </span>
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A characterization of n-associative, monotone, idempotent functions on an interval that have neutral elements

Gergely Kiss, Gábor Somlai
<span title="2017-05-25">2017</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ealgne43xjhvfetoaem5md3khq" style="color: black;">Semigroup Forum</a> </i> &nbsp;
We also present a complete characterization of idempotent monotone n-associative functions on an interval that have neutral elements. 1  ...  We investigate monotone idempotent n-ary semigroups and provide a generalisation of the Czogala-Drewniak Theorem, which describes the idempotent monotone associative functions having a neutral element.  ...  One of the most important observations of the referee was that most of the results of Section 3 can be stated in a more general setting.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00233-017-9876-3">doi:10.1007/s00233-017-9876-3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fgxtdrtprbfntadinchqljnwz4">fatcat:fgxtdrtprbfntadinchqljnwz4</a> </span>
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Idempotent functional analysis: an algebraic approach [article]

Grigory Litvinov, Viktor Maslov, Grigory Shpiz
<span title="2001-07-31">2001</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We study idempotent analogs of the main theorems of linear functional analysis and results concerning the general form of a linear functional and scalar products in idempotent spaces.  ...  We give a review of the basic ideas of Idempotent Analysis.  ...  Sobolevskiȋ who read the text carefully and made a number of corrections.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0009128v2">arXiv:math/0009128v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6geob263bfcvtobvyv73jox75m">fatcat:6geob263bfcvtobvyv73jox75m</a> </span>
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Contractive idempotents on locally compact quantum groups [article]

Matthias Neufang, Pekka Salmi, Adam Skalski, Nico Spronk
<span title="2013-03-07">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
A general form of contractive idempotent functionals on coamenable locally compact quantum groups is obtained, generalising the result of Greenleaf on contractive measures on locally compact groups.  ...  The image of a convolution operator associated to a contractive idempotent is shown to be a ternary ring of operators.  ...  The work on this article was completed during the Research in Pairs visit to the Mathematisches Forschungsinstitut Oberwolfach in August 2012.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1209.6508v2">arXiv:1209.6508v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/xn76me3lz5ax7nja7sl52ijgaa">fatcat:xn76me3lz5ax7nja7sl52ijgaa</a> </span>
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Associative idempotent nondecreasing functions are reducible

Gergely Kiss, Gábor Somlai
<span title="2018-10-17">2018</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ealgne43xjhvfetoaem5md3khq" style="color: black;">Semigroup Forum</a> </i> &nbsp;
We summarize known results when the function is defined on a chain and is nondecreasing. Our main result shows that associative idempotent and nondecreasing functions are uniquely reducible.  ...  An n-ary associative function is called reducible if it can be written as a composition of a binary associative function.  ...  Acknowledgement The authors would like to thank the referee for the valuable comments and suggestions which improved the quality of this paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00233-018-9973-y">doi:10.1007/s00233-018-9973-y</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zeysbetptfeenbj43k4uypcptm">fatcat:zeysbetptfeenbj43k4uypcptm</a> </span>
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Contractive idempotents on locally compact quantum groups

Matthias Neufang, Pekka Salmi, Adam Skalski, Nico Spronk
<span title="">2013</span> <i title="Indiana University Mathematics Journal"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ataxa6h7hzfhve6uqsuqap2ika" style="color: black;">Indiana University Mathematics Journal</a> </i> &nbsp;
The work on this article was completed during the Research in Pairs visit to the Mathematisches Forschungsinstitut Oberwolfach in August 2012.  ...  We also thank the Finnish Academy of Science and Letters, Vilho, Yrjö and Kalle Väisälä Foundation, for supporting a visit of AS to the University of Oulu.  ...  These are idempotent functionals on C 0 (G), the algebra of continuous functions on a locally compact quantum group G, and as such can be viewed as quantum counterparts of contractive idempotent measures  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1512/iumj.2013.62.5178">doi:10.1512/iumj.2013.62.5178</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zzojn7vkjrbdraxbif3goglxq4">fatcat:zzojn7vkjrbdraxbif3goglxq4</a> </span>
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Associative idempotent nondecreasing functions are reducible [article]

Gergely Kiss, Gábor Somlai
<span title="2018-09-04">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We summarize known results when the function is defined on a chain and is nondecreasing. Our main result shows that associative idempotent and nondecreasing functions are uniquely reducible.  ...  An n-ary associative function is called reducible if it can be written as a composition of a binary associative function.  ...  In Section 3.2 we complete the study of reducibility of quasitrivial, nondecreasing, associative n-ary functions (without the assumption of symmetry).  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1707.04341v3">arXiv:1707.04341v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6wme5es4ybh5dg7vqtpgl66dmu">fatcat:6wme5es4ybh5dg7vqtpgl66dmu</a> </span>
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Tropical Mathematics, Idempotent Analysis, Classical Mechanics and Geometry [article]

Grigory L. Litvinov
<span title="2010-05-07">2010</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
A brief introduction to tropical and idempotent mathematics (with an emphasys on idempotent functional analysis) is presented. Applications to classical mechanics and geometry are especially examined.  ...  The author is sincerely grateful to V. N. Kolokoltsov  ...  Let K be a b-complete semifield and A be an idempotent b-space over K equipped with the structure of a semiring compatible with the multiplication K × A → A so that the associativity of the multiplication  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1005.1247v1">arXiv:1005.1247v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nrwnfimvfzdvpoch3is2jwd4ua">fatcat:nrwnfimvfzdvpoch3is2jwd4ua</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-1005.1247/1005.1247.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> File Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/31/05/31050a2237523d4fab7159609b40805dd48b9c13.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1005.1247v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Generalization of Czogała-Drewniak Theorem for n-ary Semigroups [chapter]

Gergely Kiss, Gabor Somlai
<span title="2017-05-19">2017</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/kax3wwzwmncwhi472pxbzqsjja" style="color: black;">Advances in Intelligent Systems and Computing</a> </i> &nbsp;
We show that if F n is idempotent, n-associative function which is monotone in each of its variables, defined on an interval I ⊂ R and has a neutral element, then F n is combination of the minimum and  ...  We investigate n-ary semigroups as a natural generalization of binary semigroups. We refer it as a pair (X, F n ), where X is a set and an n-associative function F n : X n → X is defined on X.  ...  Acknowledgements The results based on the article [7] , which is an extended version of the current paper.  ... 
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Tensor products of idempotent semimodules. An algebraic approach [article]

Grigori Litvinov, Viktor Maslov, Grigori Shpiz
<span title="2001-01-18">2001</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
This is one of a series of papers on idempotent functional analysis.  ...  We study idempotent analogs of topological tensor products in the sense of A. Grothendieck. The basic concepts and results are simulated on the algebraic level.  ...  The present paper is one of a series of papers on idempotent functional analysis, which is an abstract version of idempotent analysis in the sense of [2] [3] [4] [5] [6] [7] [8] [9] .  ... 
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