TY - JOUR
TI - Minimal Subvarieties of Involutive Residuated Lattices
AU - Souma, Daisuke
TI - Minimal Subvarieties of Involutive Residuated Lattices
AB - It is known that classical logic CL is the single maximal consistent logic over intuitionistic logic Int, which is moreover the single one even over the substructural logic FLew. On the other hand, if we consider maximal consistent logics over a weaker logic, there may be uncountablymany of them. Since the subvariety lattice of a given variety V of residuated lattices is dually isomorphic to the lattice of logics over the corresponding substructural logic L(V), the number of maximal consistent logics is equal to the number of minimal subvarieties of the subvariety lattice of V. Tsinakis and Wille have shown that there exist uncountably many atoms in the subvariety lattice of the variety of involutive residuated lattices. In the present paper, we will show that while there exist uncountably many atoms in the subvariety lattice of the variety of bounded representable involutive residuated lattices with mingle axiom x2 ≤ x, only two atoms exist in the subvariety lattice of the variety of bounded representable involutive residuated lattices with the idempotency x = x2.
VL - 2011
IS - Number 46
PY - 2011
SN - 0137-2904
C1 - 2084-2589
SP - 17
EP - 27
DO - 10.4467/20842589RM.11.002.0280
UR - https://ejournals.eu/en/journal/reports-on-mathematical-logic/article/minimal-subvarieties-of-involutive-residuated-lattices