Ideals in Bipolar Quantum Linear Algebra

Since bipolar quantum linear algebra (BQLA), under two operations–-addition and multiplication—demonstrates the properties of semirings, and since ideals play an important role in abstract algebra, our results are compelling for the ideals of a semiring. In this article, we investigate the characteristics of ideals, principal ideals, prime ideals, maximal ideals, and the smallest ideal containing any nonempty subset. By applying elementary real analysis, particularly the infimum, our main result is stated as follows: for any closed set I in BQLA, I is a nontrivial proper ideal if and only if there exists $c\in (0,1]$ such that $I=\left\{(-x,y)\in {\mathbb{R}}^2|cx\le y\le {\displaystyle \frac{x}{c}}\mathrm{and}x,y\ge 0\right\}$. This shows that its shape has to be symmetric with the graph $y=-x$.