# Relation between Sheffer Stroke and Hilbert algebras

Document Type: Research Paper

Authors

1 Department of Mathematics, Ege University, 35100 Izmir, Turkey

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

Abstract

In this paper, we introduce a Sheffer stroke Hilbert algebra by giving definitions of Sheffer stroke and a Hilbert algebra. After it is shown that the axioms of Sheffer stroke Hilbert algebra are independent, it is given some properties of this algebraic structure. Then it is stated the relationship between Sheffer stroke Hilbert algebra and Hilbert algebra by defining a unary operation on Sheffer stroke Hilbert algebra. Also, it is presented deductive system and ideal of this algebraic structure. It is defined an ideal generated by a subset of a Sheffer stroke Hilbert algebra, and it is constructed a new ideal of this algebra by adding an element of this algebra to its ideal.

Keywords

### References

[1] Abbott, J.C., Implicational algebras, Bull. Math. Soc. Sci. Math. Roumanie 11(1)
(1967), 3-23 .
[2] Chajda, I., Sheffer operation in ortholattices, AActa Univ. Palack. Olomuc. Fac.
Rerum Natur. Math. 44(1) (2005), 19-23.
[3] Cornish, W.H., On positive implicative BCK-algebras, Mathematics Seminar Notes
8 (1980), 455-468.
[4] Diego, A., Sur les algebras de Hilbert, Ed. Hermann, Collection de Logique Math.
21, (1966).
[5] Henkin, L., An algebraic characterization of quantifiers, Fund. Math. 37 (1950),
63-74.
[6] Idziak, P.M., Lattice operations in BCK-algebras, Sci. Math. Jpn. 29 (1984) 839-846.
[7] Iorgulescu, A., "Algebras of logic as BCK algebras", Editura ASE, Bucharest, 2008.
[8] Iseeki, K. and Tanaka, S., An introduction to the theory of BCK-algebras, Sci. Math.
Jpn., 23(1) (1978), 1-26.
[9] Jun, Y.B., Commutative Hilbert Algebra, Soochow Journal of Mathematics, 22(4)
(1996), 477-484.
[10] McCune, W., Veroff, R., Fitelson, B., Harris, K., Feist, A., and Wos, L., Short single
axioms for Boolean algebra, J. Automat. Reason., 29(1) (2002), 1-16.
[11] Oner T., and Senturk I.,The Sheffer Stroke Operation Reducts of Basic Algebras,
Open Math. 15 (2017), 926-935.
[12] Rasiowa, H., "An algebraic approach to non-classical logics", Studies in Logic and
the Foundations of Mathematics 78, North-Holland and PWN, 1974.
[13] Sheffer, H.M., A set of five independent postulates for Boolean algebras, with appli-
cation to logical constants, Trans. Amer. Math. Soc., 14(4) (1913), 481-488.
[14] Schmid, J., Distributive lattices and rings of quotients, Coll. Math. Societatis Janos
Bolyai 33 (1980), 675-696.