There is a review of the latest results about semigroups of total transformations, of partial transformations, and of multiple-valued transformations of sets. The binary relations of order, quasi-order and equivalence were considered in context of isotone transformations. The authors mention binary relations which preserve a partial order and equivalence simultaneously, and describe different approaches of definitions of isotone partial and multiple-valued transformations. All the results are widely applied in the theory of measurements for estimation of social and economic indices.
Keywords: measurement theory; binary relation; partially ordered set; quasi-ordered set; regular semigroup; chain; antichain; partial transformation.
References
- Amerbaev V. M., Kozhukhov I. B., Revyakin A. M. Predstavleniya binarnykh otnoshenii v teorii izmerenii i poleznosti dlya otsenki sotsial’nogo kapitala (Binary Relations Representation in Measurement and Utility Theory for Social Capital Estimation), Vestnik Moskovskogo gorodskogo pedagogicheskogo universiteta, Seriya Ekonomika, 2012, No. 4 (16), pp. 66—71.
- Predstavleniya binarnykh otnoshenii i regulyarnye polugruppy izotonnykh preobrazovanii (Binary Relations Representation and Regular Semigroups of Isotone Transformations) by V. M. Amerbaev, I. B. Kozhukhov, A. M. Revyakin, V. A. Yaroshevich, Kombinatorni konfiguracii ta ih zastosuvannja, Materialy 13-go Mizhvuzivskogo naukovo-praktychnogo seminara (13—14 kvitnja 2012 r.), Kirovograd, DLAU, 2012, pp. 14—21.
- Matematicheskie metody prinyatiya reshenii i modeli v upravlenii i vo vneshneekonomicheskoi deyatel’nosti (Mathematical Methods of Decision-Making in Management and in Foreign Economic Activity) by I. N. Abanina, V. M. Amerbaev, V. V. Bardushkin i dr., pod obshch. red. I. N. Abaninoi, A. M. Revyakina, M., MGADA, 2013, 163 p.
- Yaroshevich V. A. O regulyarnykh polugruppakh izotonnykh preobrazovanii chastichno uporyadochennykh mnozhestv (On Regular Semigroups of Isotone Transformations of Partially Ordered Sets), Vestnik Moskovskogo gorodskogo pedagogicheskogo universiteta, Seriya Ekonomika, 2012, No. 4 (16), pp. 130—134.
- Klifford A., Preston G. Algebraicheskaya teoriya polugrupp (The Algebraic Theory of Semigroups), T. 1, 2, M., Mir, 1972.
- Aizenshtat A. Ya. Regulyarnye polugruppy endomorfi zmov uporyadochennykh mnozhestv (Regular Semigroups of Ordered Sets’ Endomorphism), Uchenye zapiski Leningradskogo gos. ped. in-ta im. A. I. Gertsena, 1968, T. 387, pp. 3—11.
- Adams M. E., Gould M. Posets whose monoids of order-preserving maps are regular, Order, 1989, 6 (2), pp. 195—201.
- Kim V. I., Kozhukhov I. B. Usloviya regulyarnosti polugrupp izotonnykh preobrazovanii schetnykh tsepei (Countable Chains Isotone Transformations Semigroups Regularity Conditions), Fundamental’naya i prikladnaya matematika, 2006, T. 12, Vyp. 8, pp. 97—104.
- Yaroshevich V. A. O svoistvakh polugrupp chastichnykh izotonnykh preobrazovanii kvaziuporyadochennykh mnozhestv (On Quasi-Ordered Sets’ Partial Isotone Transformations Semigroups’ Properties), Vestnik MGADA, Seriya Filosofskie, sotsial’nye i estestvennye nauki, 2011, No. 3 (9), pp. 139—144.
- Yaroshevich V. A. Otobrazheniya, soglasuyushchiesya s binarnymi otnosheniyami (Mappings Congruent to Binary Relations), Matematicheskii vestnik pedvuzov i universitetov Volgo-Vyatskogo regiona, 2009, No. 11, pp. 135—142.
- Kim V. I., Kozhukhov I. B., Yaroshevich V. A. Slabo regulyarnye polugruppy izotonnykh preobrazovanii (Weakly Regular Semigroups of Isotone Transformations), Fundamental’naya i prikladnaya matematika, 2012, T. 17, Vyp. 4, pp. 145—166.
- Huisheng Pei, Dingyu Zou. Green’s Equivalences on Semigroups of Transformations Preserving Order and an Equivalence Relation, Semigroup Forum, 2005, 71 (2), pp. 241—251.
- Lei Sun, Hui Sheng Pei. Green’s Relations on Semigroups of Transformations Preserving Two Equivalence Relations, Journal of Mathematical Research & Exposition, 2009, 29, 3, pp. 415—422.
- Deng Lun-Zhi, Zeng Ji-Wen, You Tai-Jie. Green’s relations and regularity for semigroups of transformations that preserve order and a double direction equivalence, Semigroup Forum, 2012, 84 (1), pp. 59—68.
- Kozhukhov I. B., Yaroshevich V. A. Polugruppy otobrazhenii, sokhranyayushchikh binarnoe otnoshenie (Semigroups of Mappings Preserving Binary Relation), Fundamental’naya i prikladnaya matematika, 2008, T. 14, Vyp. 7, pp. 129—136.
- Tvorogov A. V., Yaroshevich V. A. O regulyarnosti polugruppy mnogoznachnykh preobrazovanii, sokhranyayushchikh zadannoe binarnoe otnoshenie (On Regularity of Semigroup of Multiple-Valued Transformations that Preserve Given Binary Relation), Algebra, teoriya chisel i diskretnaya geometriya: sovremennye problemy i prilozheniya, Mat-ly XIII Mezhdunar. konf., posvyashch. 85-letiyu so dnya rozhd. prof. S. S. Ryshkova, Tula, Izd-vo TulGPU im. L. N. Tolstogo, 2015, pp .135—137.