Ternary hemirings and ternary semirings: definitions and examples

In this paper we mainly introduce the some special notions of ternary semi rings and we gave examples of definitions and we prove that a set T containing two distinct elements 0 and 1 and on which operations + and [ ] are defined is a commutative ternary semi ring if and only if the following conditions are satisfied for all a, b, c, d, e, f ∈ T: (1) a + 0 = 0 + a = a; (2) a11 = a; (3) 00a =0; (4) [(aeg+ b)+ c]df = dfb+ [ae(gdf) + cdf].