Sheffer stroke R0algebras

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Arts and Sciences, Izmir University of Economics, Balcova, Izmir, Turkiye.

2 Department of Mathematics, Faculty of Science, Ege University, Bornova, Izmir, Turkiye.

3 Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran.

Abstract

The main objective of this study is to introduce Sheffer stroke R0algebra (for short, SR0 algebra). Then it is stated that the axiom system of a Sheffer stroke R0algebra is independent. It is indicated that every Sheffer stroke R0algebra is R0algebra but specific conditions are necessarily for the inverse. Afterward, various ideals of a Sheffer stroke R0algebra are defined, a congruence relation on a Sheffer stroke R0algebra is determined by the ideal and quotient Sheffer stroke R0algebra is built via this congruence relation. It is proved that quotient Sheffer stroke R0algebra constructed by a prime ideal of this algebra is totally ordered and the cardinality is less than or equals to 2. After all, important conclusions are obtained for totally ordered Sheffer stroke R0algebras by applying various properties of prime ideals.

Keywords


[1] I. Chajda, Sheffer operation in ortholattices, Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math., 44 No. 1 (2005) 19-23.
[2] I. Chajda, R. Halaš and H. Länger, Operations and structures derived from non-associative MV-algebras, Soft Comput., 23 No. 12 (2019) 3935-3944.
[3] F. Esteva and L. Godo, Monoidal t-norm based logic: towards a logic for left-continous t- norms, Fuzzy Sets Syst., 124 No. 3 (2001) 271-288.
[4] P. Hájek, Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, 1998.
[5] Y. B. Jun and L. Liu, Filters of R0−algebras, Int. J. Math. Math. Sci., 2006 (2006) Article ID 93249.
[6] W. McCune, R. Veroff, B. Fitelson, K. Harris, A. Feist and L.Wos, Short single axioms for Boolean algebra, J. Autom. Reason., 29 No. 1 (2002) 1-16.
[7] A. Molkhasi and K. P. Shum, Representations of strongly algebraically closed algebras, Algebra Discrete Math., 28 No. 1 (2019) 130-143.
[8] A. Molkhasi, Representations of Sheffer stroke algebras and Visser algebras, Soft Comput., 25 (2021) 8533-8538.
[9] T. Oner, T. Katican and A. Borumand Saeid, (Fuzzy) filters of Sheffer stroke BL-algebras, Kragujev. J. Math., 47 No. 1 (2023) 39-55.
[10] T. Oner, T. Katican, A. Borumand Saeid and M. Terziler, Filters of strong Sheffer stroke non-associative MV-algebras, Analele Stiint. ale Univ. Ovidius Constanta Ser. Mat., 29 No. 1 (2021) 143-164.
[11] T. Oner, T. Katican and A. Borumand Saeid, Relation between Sheffer stroke and Hilbert Algebras, Categ. Gen. Algebr. Struct., 14 No. 1 (2021) 245-268.
[12] T. Oner, T. Katican and A. Borumand Saeid, On Sheffer stroke UP-algebras, Discuss. Math. - Gen. Algebra Appl., 41 (2021) 381-394.
[13] T. Oner, T. Katican and A. Borumand Saeid, Fuzzy filters of Sheffer stroke Hilbert algebras, J. Intell. Fuzzy Syst., 4081 (2021) 759-772.
[14] D. W. Pei and G. J. Wang, The completeness and application of formal systems L, Sci. China Series E, 32 No. 1 (2002) 56-64.
[15] D. Pei and G. Wang, The completeness and applications of the formal system L, Sci. China Inf. Sci., 45 (2002) 40-50.
[16] H. M. Sheffer, A set of five independent postulates for Boolean algebras, with application to logical constants, Trans. Am. Math. Soc., 14 No. 4 (1913) 481-488.
[17] E. Turunen, Boolean deductive systems of BL-algebras, Arch. Math. Log., 40 No. 6 (2001) 467-473.
[18] R. Veroff, A shortest 2-basis for Boolean algebra in terms of the Sheffer stroke, J. Autom. Reason., 31 No. 1 (2003) 1-9.
[19] G. J. Wang, Non-classical Mathematical Logic and Approximate Reasoning, Science Press, 2000.
[20] G. J. Wang, On the Logic Foundation of Fuzzy Reasoning, Inf. Sci., 117 (1999) 47-88.
[21] Y. Xu, Lattice implication algebras, J. Southwest Jiaotong Univ., 89 No. 1 (1993) 20-27.
[22] Y. Xu and K. Y. Qin, On filters of lattice implication algebras, J. fuzzy math., 1 No. 2 (1993) 251-260.