ON r-IDEALS IN INCLINE ALGEBRAS
نویسندگان
چکیده
In this paper we show that if K is an incline with multiplicative identity and I is an r-ideal of K containing a unit u, then I = K. Moreover, we show that in a non-zero incline K with multiplicative identity and zero element, every proper r-ideal in K is contained in a maximal r-ideal of K. Z. Q. Cao, K. H. Kim and F. W. Roush [3] introduced the notion of incline algebras in their book, Incline algebra and applications, and these concepts were studied by some authors ([2, 4, 5]). Inclines are a generalization of both Boolean and fuzzy algebras, and a special type of a semiring, and give a way to combine algebras and ordered structures to express the degree of intensity of binary relations. The present authors with Y. B. Jun [2] introduced the notion of quotient incline and obtained the structure of incline algebras, and also introduced the notion of prime and maximal ideals in an incline, and studied some relations between them in incline algebras. In this paper, as a continuation of [2], we show that if K is an incline with multiplicative identity and I is an r-ideal of K containing a unit u, then I = K. Moreover, we show that in a non-zero incline K with multiplicative identity and zero element, every proper r-ideal in K is contained in a maximal r-ideal of K. Definition 1 ([3]). An incline (algebra) is a set K with two binary operations denoted by “ + ” and “ ∗ ” satisfying the following axioms: for all x, y, z ∈ K, (i) x + y = y + x, (ii) x + (y + z) = (x + y) + z, (iii) x ∗ (y ∗ z) = (x ∗ y) ∗ z, (iv) x ∗ (y + z) = (x ∗ y) + (x ∗ z), Received May 24, 2001. Revised October 9, 2001. 2000 Mathematics Subject Classification: 16Y60, 16Y99.
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