State extended ideals in state MV-algebras

Subject Areas : Statistics

Fereshteh Forouzesh ¹ , zahra Dehghani Poshtroudi ² , Tayebeh Waezizadeh ³ , Mahdieh Ebrahimpour ⁴

1 - Department of Pure Mathematics, Faculty of Mathematics, Higher Education Complex of Bam, Kerman, Iran
2 - Department of Pure Mathematics, Faculty of Mathematics, Higher Education Complex of Bam, Kerman, Iran
3 - Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Iran
4 - Department of Mathematics,Vali-e-Asr University,P.O. Box 7713936417, Rafsanjan, Iran

Received: 2018-09-03 Accepted : 2021-04-27 Published : 2021-05-22

Keywords: ایده‌آل حالت توسعه یافته, زنجیر حالت, MV-جبر حالت, پایدار,

Abstract :

In this paper, we introduce the notion of state extended ideal associated to a subset of state MV-algebras ‎‎and‎‎‎‎‎‎ investigate the related properties. A characterization of this state extended ideal in a state MV-algebras is given‎. In addition, we study the relation between state extended ideals and state prime ideals, state maximal ideal in a state MV-algebra.We show that if f:A→B is state homomorphism MV-algebra such that f(A^' )=B', then we have (1) If I is a state stable relative to B'⊆B, then f^(-1) (I) is a state stable relative to A'⊆A.(2) If f is an onto and I is a state stable relative to A'⊆A, Ker(f)⊆I, then f(I) is a state stable relative to B'⊆B. ‎In ‎finally, ‎we define state stable ideals and we show that the ‎class ‎‎‎S(B)‎ of ‎all ‎state stable ‎ideals ‎relative ‎to ‎‎ B⊆A ‎is ‎also a‎ ‎complete ‎Heyting ‎algebra, ‎for ‎a state ‎‎MV‎-algebra ‎‎(A,σ).

References:

 

[1] محمد سالار بارده، ایده­الهای سرسخت n-لایه در MV-جبرها، پایان نامه ارشد دانشگاه شهید باهنر کرمان،  (1395).

[2] C. C. Chang, Algebraic analysis of many valued logic, Trans. Amer. Math. Soc., (1958), 467-490.

[3] R. Cignoli, I. M. L. D’Ottaviano, D. Mundici, Algebraic Foundations of Many-Valued Reasoning, Kluwer Academic, Dordrecht, (2000).

• [4] C. Ciungu, A. Dvurecenskij, M. Hycko, State -algebras, Soft Comput., 15 (2011), 619-634.


• [5] Forouzesh, Extended ideals in -algebras, Submitted.


• [6] Forouzesh, A. Darijani, Some classes of state ideals in state -algebras, Eurasian Mathematical journal,10 (2019), 37-48.


• [7] Forouzesh, State radical of state ideals in state -algebras, Submitted.


• [8] Flaminio, F. Montagna, An algebraic approach to states on -algebras. In: Novak V (ed) Fuzzy Logic 2, proceedings of the 5^th EUSFLAT conference, September 1114, Ostrava, Vol II (2007), pp 201-206.


• [9] Flaminio, F. Montagna, -algebras with internal states and probabilistic fuzzy logic. Int J Approx Reason, 50 (2009), 138-152.


• Georgescu, Bosbach states on fuzzy structures. Soft Comput, 8 (2004), 217-230.


• Iorgulescu, Algebras of logic as -algebras, Academy of economic studies Bucharest, Romania, (2008).


• Kroupa, Every state on semisimple -algebra is integral. Fuzzy Sets Syst, 157 (2006), 2771-2782.


• Kuhr, D. Mundici, De Fineti theorem and Borel states in -valued algebraic logic. Int J Approx Reason, 46 (1986), 15-63.


• Mundici, Averaging the truth value in Lukasiewicz sentential logic. Studia Logica, 55 (1995), 113-127.


• Piciu. Algebras of fuzzy logic, Ed. Universitaria Craiova (2007).


• Rachunek, D. Salounova, State operators on -algebras, Soft Comput, 15 (2011), 327-334.