SHARP ELEMENTS IN d0-ALGEBRAS

Document Type : Research Paper

Authors

University of Basilicata, Department of Mathematics, Computer Science and Economics

Abstract

We introduce the notions of Sasaki mapping and of sharp elements on a d0-algebra. We investigate the relationships of sharp elements with Sasaki mappings and with central elements, thus generalizing some results known for D-lattices. Namely, we give a characterization of sharp elements, by means of Sasaki mappings, which extends a result of Bennett and Foulis; we also prove that an element is central if and only if it is a sharp element and every element is compatible with it: this generalizes a result of Riecanová.

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References

[1] A. Avallone, G. Barbieri, P. Vitolo, H. Weber, Decomposition of e ect algebras and the Hammer-Sobczyk theorem,
Algebra Universalis, 60(1) (2009), 1-18.
[2] A. Avallone, A. de Simone, P. Vitolo, E ect algebras and extensions of measures, Bollettino dell’Unione Matematica
Italiana, 9(2) (2006), 423-444.
[3] A. Avallone, P. Vitolo, Hahn decomposition of modular measures and applications, Annales Soc. Math. Polon.
Series I: Commentationes Mathematicae, 43 (2003), 149-168.
[4] A. Avallone, P. Vitolo, Decomposition and control theorems on e ect algebras, Scientiae Mathematicae Japonicae,
58(1) (2003), 1-14.
[5] A. Avallone, P. Vitolo, Lattice uniformities on e ect algebras, International Journal of Theoretical Physics, 44(7)
(2005), 793-806.
[6] A. Avallone, P. Vitolo, Lyapunov decomposition of measures on e ect algebras, Scientiae Mathematicae Japonicae,
69(1) (2009), 79-87.
[7] A. Avallone, P. Vitolo, Hahn decomposition in d0-algebras, Soft Computing, 23(22) (2019), 11373-11388.
[8] A. Avallone, P. Vitolo, Lyapunov decomposition in d0-algebras, Rendiconti del Circolo Matematico di Palermo
Series 2, 69 (2020), 837-859.
[9] A. Avallone, P. Vitolo, The center of a d0-algebra, Reports on Mathematical Physics, 86(1) (2020), 63-78.
[10] A. Avallone, P. Vitolo, Kalmbach Measurability in d0-algebras, Mathematica Slovaca, 72(6) (2022), 1387-1402.
[11] A. Avallone, P. Vitolo, Decomposition of d0-algebras, Submitted for publication.
[12] M. K. Bennett, D. J. Foulis, E ect algebras and unsharp quantum logics, Foundations of Physics, 24(10) (1994),
1331-1352.
[13] M. K. Bennett, D. J. Foulis, Phi-symmetric e ect algebras, Foundations of Physics, 25(12) (1994), 1699-1722.
[14] F. Chovanec, F. Kˆopka, D-lattices, International Journal of Theoretical Physics, 34 (1995), 1297-1302.
[15] C. Constantinescu, Some properties of spaces of measures, Atti del Seminario Matematico e Fisico dell’ Universita
di Modena, 35(suppl.), Modena, Italy, 1989.
[16] A. Dvureˇcenskij, States on weak pseudo EMV-algebras. I. States and states morphisms, Iranian Journal of Fuzzy
Systems, 19(4) (2022), 1-15.
[17] A. Dvureˇcenskij, States on weak pseudo EMV-algebras. II. Representations of states, Iranian Journal of Fuzzy
Systems, 19(4) (2022), 17-26.
[18] A. Dvureˇcenskij, M. G. Graziano, Remarks on representations of minimal clans, Tatra Mountains Mathematical
Publications, 15 (1998), 31-53.
[19] A. Dvureˇcenskij, M. G. Graziano, On representations of commutative BCK-algebras, Demonstratio Mathematica,
32(2) (1999), 227–246.
[20] A. Dvureˇcenskij, S. Pulmannov´a, New trends in quantum structures, Kluwer Academic Publishers, Dordrecht,
2000.
[21] D. J. Foulis, A note on orthomodular lattices, Portugaliae Mathematica, 21(1) (1962), 65-72.
[22] J. J. M. Gabri¨els, S. M. Gagola, M. Navara, Sasaki projections, Algebra Universalis, 77(3) (2017), 305-320.
[23] M. Nakamura, The permutability in a certain orthocomplemented lattice, Kodai Mathematical Seminar Reports,
9(4) (1957), 158-160.
[24] Z. Rieˇcanov´a, Compatibility and central elements in e ect algebras, Tatra Mountains Mathematical Publications,
16 (1999), 151-158.
[25] Z. Rieˇcanov´a, Subalgebras, intervals and central elements of generalized e ect algebras, International Journal of
Theoretical Physics, 38 (1999), 3209-3220.
[26] M. Rosa, P. Vitolo, Blocks and compatibility in d0-algebras, Algebra Universalis, 78(4) (2017), 489-513.
[27] M. Rosa, P. Vitolo, Topologies and uniformities on d0-algebras, Mathematica Slovaca, 67(6) (2017), 1301-1322.
[28] M. Rosa, P. Vitolo, Measures and submeasures on d0-algebras, Ricerche di Matematica, 67 (2018), 373-386.
[29] U. Sasaki, Orthocomplemented lattices satisfying the exchange axiom, Journal of Science of the Hiroshima University,
Series A, 17 (1954), 293-302.
[30] K. D. Schmidt, Minimal clans: A class of ordered partial semigroups including Boolean rings and lattice-ordered
groups, in Semigroups, theory and applications (Oberwolfach, 1986), volume 1320 of Lecture Notes in Math.,
Springer, Berlin, (1988), 300-341.
[31] K. D. Schmidt, Jordan decompositions of generalized vector measures, volume 214 of Pitman Research Notes in
Mathematics Series, Longman Scientific and Technical, Harlow, 1989.
[32] P. Vitolo, A generalization of set-di erence, Mathematica Slovaca, 61(6) (2011), 835-850.
[33] H. Weber, An abstraction of clans of fuzzy sets, Ricerche di Matematica, 46(2) (1997), 457-472.
[34] O. Wyler, Clans, Compositio Mathematica, 17 (1965), 172-189.
Iranian Journal of Fuzzy Systems
Volume 20, Issue 6
November and December 2023
Pages 85-103

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