Algebras on subintervals of BL-algebras, pseudo BL-algebras and bounded residuated $\ell $-monoids
Anatolij Dvurečenskij, Marek Hyčko · Czech digital mathematics library · 2006
Let M be a BL-algebra or a pseudo BL-algebra or a bounded residuated ^-monoid and let a < b, a, b G 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.-KUHR, J.: A note on interval MV-algebras, Math.Slovaca 56 (2006), 47-52] and [CHAJDA, I.-KUHR, J.: GMV-algebras and meet-semilattices with sectionally antitone "permutations, Math.Slovaca (To appear)], [JAKUBIK, 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.