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Some properties of state filters in state residuated lattices

We consider properties of state filters of state residuated lattices and prove that for every state filter F of a state residuated lattice X: (1) F is obstinate ⇔ L/F ∼= {0, 1}; (2) F is primary ⇔ L/F is a state local residuated lattice; and that every g-state residuated lattice X is a subdirect product of {X/Pλ}, where Pλ is a prime state filter of X. Moreover, we show that the quotient MTL-algebra X/P of a state residuated lattice X by a state prime filter P is not always totally ordered, although the quotient MTL-algebra by a prime filter is totally ordered.