YES

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
a(x) -> x
a(a(x)) -> b(c(x))
b(x) -> x
c(x) -> x
c(b(x)) -> b(a(c(x)))

CP(R):
a(___y654) -> b(c(___y654))
a(___y653) -> b(c(___y653))
b(c(___y661)) -> a(___y661)
a(b(c(___y663))) -> b(c(a(___y663)))
c(___y679) -> b(a(c(___y679)))
b(___y690) -> b(a(c(___y690)))
b(a(c(___y697))) -> b(___y697)

critical-pair-closing system C:
a(x) -> x
b(x) -> x
c(x) -> x

CR(C): see below.

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
a(x) -> x
b(x) -> x
c(x) -> x

CP(R):


critical-pair-closing system C:


CR(C): see below.

---- CR(R)
---- R is empty.