## Reports on Mathematical Logic

### No. 37

Hiroki TAKAMURA:

Every Free Biresiduated Lattice is Semisimple

A b s t r a c t. In this paper, we prove the semisimplicity of free biresiduated lattices, more precisely, integral residuated lattices. In [4], authors show that variety of residuated lattices, more precisely, commutative integral residuated lattices, is generated by its finite simple members. The result is obtained by showing that every {\it free} residuated lattice is semisimple and then showing that every variety generated by a simple residuated lattice is generated by a set of finite simple residuated lattices. The proof of the former is based on Gri\v{s}in's idea in [2]. We show that their proof of the semisimplicity works well also for free biresiduated lattices.