We introduce several restricted  versions of 

 Gentzen's structural and logical rules,

 and  investigate cut-elimination property, 

theorem-equivalence, Ackermann's  property, 

decidability and variable sharing property 

among  implicational sequent calculi  

having the  rules.

These results include new cut-elimination 

theorems for the implicational fragments 

of the following:

relevant logic E of entailment, EW,  

strict implication S4,

S4W  and   full Lambek logic FL.

Next  we give Kripke type semantics 

for the implicational  fragments of  EW, 

S4W and related logics (e.g.,  BCK, BCI and BB$'$I).

Further we prove the completeness theorems 

for the semantics by using Ishihara's canonical 

model construction method.