TITLE:
A Class of Semimodules with a Free Basis of Cardinality 2
AUTHORS:
Jun Li, Xingjun Zhang
KEYWORDS:
Semiring, Quotient Semirings, Free Semimodules, Free Basis
JOURNAL NAME:
Open Journal of Applied Sciences,
Vol.16 No.9,
September
18,
2026
ABSTRACT: A class of semimodules with a free basis of cardinality 2 is investigated. We present the semiring form for the existence of a basis of cardinality 1 in such semimodules, and construct semirings satisfying the condition by means of congruences. In addition, we discuss the freeness of bases of cardinality 1, and prove that
{
(
c,d
)
T
}
forms a free basis of
V
2
(
S
)
if and only if
ca+db=1
.