Residuation in orthomodular lattices

Residuation in orthomodular lattices

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We show that every idempotent weakly divisible residuated lattice satisfying the double negation ALL PURPOSE CL law can be transformed into an orthomodular lattice.The converse holds if adjointness is replaced by conditional adjointness.Moreover, we show that every positive right residuated lattice satisfying the double negation law and Body Wash two further simple identities can be converted into an orthomodular lattice.

In this case, also the converse statement is true and the corresponence is nearly one-to-one.