Super-De Morgan functions and free De Morgan quasilattices

Yuri Movsisyan; Vahagn Aslanyan

Open Mathematics (2014)

  • Volume: 12, Issue: 12, page 1749-1761
  • ISSN: 2391-5455

Abstract

top
A De Morgan quasilattice is an algebra satisfying hyperidentities of the variety of De Morgan algebras (lattices). In this paper we give a functional representation of the free n-generated De Morgan quasilattice with two binary and one unary operations. Namely, we define the concept of super-De Morgan function and prove that the free De Morgan quasilattice with two binary and one unary operations on nfree generators is isomorphic to the De Morgan quasilattice of super-De Morgan functions of nvariables.

How to cite

top

Yuri Movsisyan, and Vahagn Aslanyan. "Super-De Morgan functions and free De Morgan quasilattices." Open Mathematics 12.12 (2014): 1749-1761. <http://eudml.org/doc/268963>.

@article{YuriMovsisyan2014,
abstract = {A De Morgan quasilattice is an algebra satisfying hyperidentities of the variety of De Morgan algebras (lattices). In this paper we give a functional representation of the free n-generated De Morgan quasilattice with two binary and one unary operations. Namely, we define the concept of super-De Morgan function and prove that the free De Morgan quasilattice with two binary and one unary operations on nfree generators is isomorphic to the De Morgan quasilattice of super-De Morgan functions of nvariables.},
author = {Yuri Movsisyan, Vahagn Aslanyan},
journal = {Open Mathematics},
keywords = {Antichain; De Morgan algebra; Hyperidentity; Hypervariety; De Morgan quasilattice; De Morgan function; Subdirectly irreducible algebra; Free algebra; Super-De Morgan function; Hyper-De Morgan function; Disjunctive normal form (DNF) of super-De Morgan function; antichain; hyperidentity; hypervariety; subdirectly irreducible algebra; free algebra; super-De Morgan function; hyper-De Morgan function; disjunctive normal form (DNF) of super-De Morgan function},
language = {eng},
number = {12},
pages = {1749-1761},
title = {Super-De Morgan functions and free De Morgan quasilattices},
url = {http://eudml.org/doc/268963},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Yuri Movsisyan
AU - Vahagn Aslanyan
TI - Super-De Morgan functions and free De Morgan quasilattices
JO - Open Mathematics
PY - 2014
VL - 12
IS - 12
SP - 1749
EP - 1761
AB - A De Morgan quasilattice is an algebra satisfying hyperidentities of the variety of De Morgan algebras (lattices). In this paper we give a functional representation of the free n-generated De Morgan quasilattice with two binary and one unary operations. Namely, we define the concept of super-De Morgan function and prove that the free De Morgan quasilattice with two binary and one unary operations on nfree generators is isomorphic to the De Morgan quasilattice of super-De Morgan functions of nvariables.
LA - eng
KW - Antichain; De Morgan algebra; Hyperidentity; Hypervariety; De Morgan quasilattice; De Morgan function; Subdirectly irreducible algebra; Free algebra; Super-De Morgan function; Hyper-De Morgan function; Disjunctive normal form (DNF) of super-De Morgan function; antichain; hyperidentity; hypervariety; subdirectly irreducible algebra; free algebra; super-De Morgan function; hyper-De Morgan function; disjunctive normal form (DNF) of super-De Morgan function
UR - http://eudml.org/doc/268963
ER -

References

top
  1. [1] Anosov A.D., On homomorphisms of many-sorted algebraic systems in connection with cryptographic applications, Discrete Math. Appl., 2007, 17(4), 331–347. http://dx.doi.org/10.1515/dma.2007.028 Zbl1282.08005
  2. [2] Arieli O., Avron A., The value of four values, Artificial Intelligence, 1998, 102, 97–141. http://dx.doi.org/10.1016/S0004-3702(98)00032-0 Zbl0928.03025
  3. [3] Balbes R., Dwinger P., Distributive lattices, Univ. of Missouri Press, 1974. Zbl0321.06012
  4. [4] Belnap N.D., A useful four valued logic, in: G. Epstein, J.M. Dunn (Eds)., Modern Uses of Multiple-Valued Logic, Reidel Publishing Comnpany, Boston, 1977, 7–73. 
  5. [5] Belousov V.D., Systems of quasigroups with generalized identities, Uspekhi Mat. Nauk, 1965, 20, 75–144. English transl. in Russian Math. Surveys 1965, 20, 73–143. Zbl0135.03503
  6. [6] Berman J., Blok W., Stipulations, multi-valued logic and De Morgan algebras, Multi-valued Logic 2001, 7(5–6), 391–416. Zbl1016.03068
  7. [7] Birkhoff G., Lattice Theory. 3rd edn., American Mathematical Society, Providence, Rhode Island, 1967. 
  8. [8] Bou F., Rivieccio U., The logic of distributive bilattices, Log. J. IGPL, 2011, 19, 183–216. http://dx.doi.org/10.1093/jigpal/jzq041 Zbl1214.03056
  9. [9] Brzozowski J.A., A characterization of De Morgan algebras, International Journal of Algebra and Computation, 2001, 11, 525–527. http://dx.doi.org/10.1142/S0218196701000681 Zbl1025.06007
  10. [10] Brzozowski J.A., De Morgan bisemilattices, Proceedings of the 30th IEEE International Symposium on Multiple-Valued Logic (ISMVL 2000), May 23–25, 2000, p.173. 
  11. [11] Brzozowski J.A., Partially ordered structures for hazard detection, Special Session: The Many Lives of Lattice Theory, Joint Mathematics Meetings, San Diego, CA, January 6–9, 2002. 
  12. [12] Church A., Introduction to Mathematical Logic, Vol.1, Princeton University Press, Princeton, NJ 1956. (Volume 2 was newer published) Zbl0073.24301
  13. [13] Crama Y., Hammer P.L., Boolean Functions: Theory, Algorithms, and Applications, Cambridge University Press, New York, 2011. http://dx.doi.org/10.1017/CBO9780511852008 Zbl1237.06001
  14. [14] Denecke K., Wismath Sh.L., Hyperidentities and Clones, Gordon and Breach Science Publishers, 2000. Zbl0960.08001
  15. [15] Gehrke M., Walker C., Walker E., A mathematical setting for fuzzy logics, Int. Journ. of Uncertainty, Fuzziness and Knowledge-based Systems 1997, 5(3), 223–238. http://dx.doi.org/10.1142/S021848859700021X Zbl1232.03016
  16. [16] Gehrke M., Walker C., Walker E., Some comments on fuzzy normal forms, Proc. of the ninth IEEE Int. Conf. on Fuzzy Systems, FUZZ-IEEE, 2000, 7, 593–598. 
  17. [17] Grätzer G., Lattice Theory: Foundation, Springer Basel AG, 2011. Zbl1233.06001
  18. [18] Grätzer G., Universal Algebra, Springer-Verlag, 2010. 
  19. [19] Hazewinkel M. (Editor), Handbook of algebra, Vol. 2, North-Holland, 2000. 
  20. [20] Kalman J.A., Lattices with involution, Trans. Amer. Math. Soc., 1958, 87, 485–491. http://dx.doi.org/10.1090/S0002-9947-1958-0095135-X Zbl0228.06003
  21. [21] Kauffman L.H., De Morgan Algebras - completeness and recursion, Proceedings of the eighth international symposium on Multiple-valued logic, IEEE Computer Society Press Los Alamitos, CA, USA, 1978, 82–86. 
  22. [22] Kondo M., Characterization theorem of 4-valued De Morgan logic, Mem. Fac. Sci. Eng. Shimane Univ., Series B: Mathematical Science, 1998, 31, 73–80. Zbl0929.03032
  23. [23] Koppitz J., Denecke K., M-Solid Varieties of Algebras, Springer, 2006. Zbl1094.08001
  24. [24] Maltsev A.I., Algebraic systems, Grundlehren der mathematishen Wissenschaften, Vol.192, Springer-Verlag, New-York. Zbl1112.37052
  25. [25] Maltsev A.I., Some questions of the theory of classes of models, Proceedeings of the fourth All-Union Mathematics Congress, 1963, 1, 169–190 (Russian). 
  26. [26] Markov A.A., Constructive logic (in Russian), Uspekhi Mat. Nauk, 1950, 5, 187–188. 
  27. [27] Melkonian V., Circuit integrating through lattice hyperterms,Discrete Math. Algorithms Appl., 2011, 3(1), 101–119. http://dx.doi.org/10.1142/S179383091100105X Zbl1219.90145
  28. [28] Mobasher B., Pigozzi D., Slutzki G., Multi-valued logic programming semantics, An algebraic approach, Theorit. Comput. Sci., 1997, 171, 77–109. http://dx.doi.org/10.1016/S0304-3975(96)00126-0 Zbl0874.68046
  29. [29] Moisil G.C., Recherches sur l’algebre de la logique, Annales scientifiques de l’universite de Jassy, 1935, 22, 1–117. 
  30. [30] Movsisyan Yu.M., Binary representations of algebras with at most two binary operations. A Cayley theorem for distributive lattices, International Journal of Algebra and Computation, 2009, 19(1), 97–106. http://dx.doi.org/10.1142/S0218196709004993 Zbl1174.06006
  31. [31] Movsisyan Yu.M., Introduction to the theory of algebras with hyperidentities, Yerevan State University Press, Yerevan, 1986 (Russian). Zbl0675.08001
  32. [32] Movsisyan Yu.M., Hyperidentities and hypervarieties in algebras, Yerevan State University Press, Yerevan, 1990 (Russian). Zbl0728.08013
  33. [33] Movsisyan Yu.M., Bilattices and hyperidentities, Proc. Steclov Inst. Math., 2011, 274, 174–192. http://dx.doi.org/10.1134/S0081543811060113 
  34. [34] Movsisyan Yu.M., Hyperidentities of Boolean algebras, Izv. Ross. Akad. Nauk Ser.Mat., 1992, 56(3), 654–672. English transl. in Russ.Acad.Sci Izv. Math., 1993, 40, 607–622. Zbl0773.08003
  35. [35] Movsisyan Yu.M., Hyperidentities in algebras and varieties, Uspekhi Mat. Nauk, 1998, 53(1(319)), 61–114. English transl. in Russian Math. Surveys 1998, 53(1), 57–108. http://dx.doi.org/10.4213/rm9 
  36. [36] Movsisyan Yu., Hyperidentities and hypervarieties, Sci. Math. Jpn., 2001, 54, 595–640. Zbl1003.08001
  37. [37] Movsisyan Yu.M., On the representations of De Morgan algebras, Trends in logic III, Studialogica, Warsaw, 2005 http://www.ifispan.waw.pl/studialogica/Movsisyan.pdf 
  38. [38] Movsisyan, Yu.M., Boolean bisemigroups. Bigroups and local bigroups, CSIT Proceedings of the Conference, September 19–23, Yerevan, Armenia, 2005, 97–104. 
  39. [39] Movsisyan Yu.M., Interlaced, modular, distributive and Boolean bilattices, Armenian Journal of Mathematics, 2008, 1(3), 7–13. Zbl1281.06005
  40. [40] Movsisyan Yu.M., Algebras with hyperidentities of the variety of Boolean algebras, Izv. Ross. Akad. Nauk Ser.Mat., 1996, 60(6), 127–168. English transl. in Russ.Acad.Sci.Izv. Math., 1996, 60, 1219–1260. http://dx.doi.org/10.4213/im98 
  41. [41] Movsisyan Yu.M., Aslanyan V.A., Hyperidentities of De Morgan algebras, Log. J. IGPL, 2012, 20, 1153–1174 (doi:10.1093/jigpal/jzr053). http://dx.doi.org/10.1093/jigpal/jzr053 Zbl1276.06005
  42. [42] Movsisyan Yu.M., Aslanyan V.A., A functional representation of free De Morgan algebras, Proceedings of the Yerevan State University, Physical and Mathematical Sciences, 2012, 3, 14–16. Zbl1301.06028
  43. [43] Movsisyan Yu.M., Aslanyan V.A., De Morgan functions and free De Morgan algebras, Demonstratio Math. http://www.mini.pw.edu.pl/demmath/papers_2008/2012-145-1.pdf. 
  44. [44] Movsisyan Yu.M., Aslanyan V.A., On computation of De Morgan and quasi-De Morgan functions, Computer Science and Information Technologies (CSIT), 2013, 1–6. IEEE Conference Publications (DOI: 10.1109/CSITechnol.2013.6710334). Zbl06323590
  45. [45] Movsisyan Yu.M., Pashazadeh J., Matrix characterization of 4-ary algebraic operations of idempotent algebras, Comm. Algebra, 2014, 42, 2533–2541. http://dx.doi.org/10.1080/00927872.2013.824459 Zbl1303.08002
  46. [46] Movsisyan Yu.M., Aslanyan V.A., A functional completeness theorem for De Morgan functions, Discrete Appl. Math., 2014, 162, 1–16. http://dx.doi.org/10.1016/j.dam.2013.08.006. http://dx.doi.org/10.1016/j.dam.2013.08.006 Zbl06344065
  47. [47] Movsisyan Yu.M., Aslanyan V.A., Algebras with hyperidentities of the variety of De Morgan algebras, J. Contemp. Math. Anal., 2013, 5, 189–196. Zbl1302.06015
  48. [48] Movsisyan Yu.M., Aslanyan V.A., Subdirectly irreducible algebras with hyperidentities of the variety of De Morgan algebras, J. Contemp. Math. Anal., 2013, 6, 52–58. Zbl1302.06016
  49. [49] Movsisyan Yu.M., Aslanyan V.A., Boole-De Morgan algebras and quasi-De Morgan functions, Comm. Algebra (accepted). Zbl06323590
  50. [50] Movsisyan Yu.M., Aslanyan V.A., Super-Boolean functions and free Boolean quasilattices, Discrete Math. Algorithm. Appl., 2014, 6(2), 1450024 (13 pages) (DOI: 10.1142/S179380914500244). http://dx.doi.org/10.1142/S1793830914500244 
  51. [51] Movsisyan Yu.M., Romanowska A.B., Smith J.D.H., Superproducts, hyperidentities, and algebraic structures of logic programming, Comb. Math. and Comb. Comp., 2006, 58, 101–111. Zbl1108.68028
  52. [52] Sankappanavar H.P., A characterization of principal congruences of DeMorgan algebras and its applications, Math. Logic in Latin America, Proc. IV Latin Amer. Symp. Math. Logic, Santiago, (1978), 341–349. Nort-Holland Pub. Co., Amsterdam, 1980. 
  53. [53] Smith J.D.H., Romanowska A.B., Post-modern algebra, A Wiley-Interscience Publication, John Wiley and Sons, Inc., New York, 1999. http://dx.doi.org/10.1002/9781118032589 Zbl0946.00001
  54. [54] Pashazadeh J., A characterization of De Morgan bisemigroup of binary functions, International Journal of Algebra and Computation, 2008, 18, 951–956. http://dx.doi.org/10.1142/S021819670800472X Zbl1158.06005
  55. [55] Plotkin B.I., Universal algebra, algebraic logic, and databases, Kluwer Academic Publisher, 1994. http://dx.doi.org/10.1007/978-94-011-0820-1 Zbl0785.68025
  56. [56] Taylor W., Hyperidentities and hypervarieties, Aequationes Math., 1981, 23, 30–49. http://dx.doi.org/10.1007/BF02188010 Zbl0491.08009

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.