Algebras on subintervals of BL-algebras, pseudo BL-algebras and bounded residuated $\ell$-monoids

Year, pages: 2006, 125 - 144

Let $M$ be a BL-algebra or a pseudo BL-algebra or a bounded residuated $\ell$-monoid and let $a≤ b$, $a,b\in M$. We endow the subinterval $[a,b]$ with algebraic structure to form an algebra of the same kind as the original one. Obtained results generalize ones presented in [CHAJDA, I.—KÜHR, J.: A note on interval MV-algebras, Math. Slovaca 56 (2006), 47–52] and [CHAJDA, I.—KÜHR, J.: GMV-algebras and meet-semilattices with sectionally antitone permutations, Math. Slovaca (To appear)], [JAKUBÍK, J.: On interval subalgebras of generalized MV-algebras, Math. Slovaca 56 (2006) (To appear)] for MV-algebras and pseudo MV- (GMV-) algebras, respectively. We also study restrictions of Bosbach states on such subinterval algebras. We show that for a commutative case it is necessary to introduce one additional condition and for a non-commutative case it is necessary to introduce two conditions, left and right one, in order that the restriction of a state can define a state on the subinterval. We prove that BL-algebras always satisfy the additional condition.

Dvurečenskij, A., Hyčko, M. 2006. Algebras on subintervals of BL-algebras, pseudo BL-algebras and bounded residuated $\ell$-monoids. In Mathematica Slovaca, vol. 56, no.2, pp. 125-144. 0139-9918.

Dvurečenskij, A., Hyčko, M. (2006). Algebras on subintervals of BL-algebras, pseudo BL-algebras and bounded residuated $\ell$-monoids. Mathematica Slovaca, 56(2), 125-144. 0139-9918.