On K-regular Additive Ternary Semirings

We introduce the concepts of a k-regular and a k-invertible additive ternary semiring. We show that (i) If I is a k-regular ideal of an additive ternary semiring S and J is any ideal of S, then I ? J is a k-regular ideal of S; (ii) If S is an additively idempotent, commutative additive ternary semiring and x ? S, then M (x) is a commutative additive ternary monoid of (S, +); (iii) An additively idempotent additive ternary semiring S is k-regular if and only if S is k-invertible; (iv) Let S be an additively and lateral cancellative additive ternary semiring. If a, b ? S, then V (a) and V (b) are either disjoint or equal.