States on finite linearly ordered IMTL-algebras

Abstract

The aim of this paper is to study states on finite linearly ordered IMTL-algebras. We prove that Bosbach states, Riečan states and state-morphisms are equivalent on linearly ordered IMTL-algebras. Furthermore, we investigate the existence of states on finite linearly ordered IMTLalgebra and prove that if L is locally finite, then L has a state if and only if L is an MV-algebra, and that if L is peculiar and HðLÞ [ f0; 1g is a subalgebra of L, then L has a state if and only if ordðnÞ ¼ jHðLÞj þ 1, where HðLÞ ¼ fx 2 LjordðxÞ\1; ordð:xÞ\1g; n ¼ max HðLÞ.

Group Representation for Even and Odd Involutive Commutative Residuated Chains

“…In line with the tradition of representing classes of residuated lattices by simpler and better known structures such as groups or Boolean algebras, Mundici's celebrated categorical equivalence theorem represents MValgebras -a variety which corresponds to Lukasiewicz logic L [12] -by -groups with strong units using a truncation construction [43]. By dropping the divisibility axiom of MV-algebras one obtains the variety of IMTLalgebras which correspond to the logic IMTL [5,9,10,13,14,16,17,24,37,44,45]. The non-integral analogue of the class of IMTL-algebras shall be represented by o-groups in this paper.…”

State filters in state residuated lattices

“…States on M V -algebras have been deeply investigated [4,11]. Consequently, the notion of states has been extended to other logical algebras such as BL-algebras [19], M T Lalgebras [13], [15], R 0 -algebras [14], and residuated lattices [3], [20] and their non-commutative cases. Different approaches to the generalization mainly gave rise to two different notions, namely, Rie can states [19] and Bosbach states [6].…”