The variety of modal FL ew -algebras is generated by its finite simple members
Hiroki Takamura · Advances in Modal Logic · 2006
In this paper, we prove that the variety of modal FLew-algebras is generated by its finite simple members. The result is obtained by showing that every free modal FLew-algebra is semisimple and then showing that every variety generated by a simple modal FLew-algebra is generated by a set of finite simple modal FLew-algebras. In (7), the authors show that the variety of FLew-algebras is generated by its finite simple members. The result is obtained by first showing that every free FLew-algebra is semisimple and then showing that every variety generated by a simple FLew-algebra is generated by a set of finite simple FLew-algebras. To show the former, based on Griin's idea in (4) authors introduced a sequent system SFL +w such that 1. algebras for SFL +w are exactly equal to FLew-algebras, 2. cut elimination theorem holds for SFL +w. Then, using proof-theoretic properties of SFL +w, the semisimplicity of free FLew-algebras is obtained. Moreover, they show that the finite embeddabil- ity property holds for simple FLew-algebras. Finally, they have the variety of FLew-algebras is generated by its finite simple members. In this paper, their proof works well also for the variety of modal FLew- algebras with some modification. We assume a familiarity with the paper (7).