Z-ideals and Z-congruences on semiring \mathcal{R}^+(L)

For a frame L, \mathcal{R}^+(L) denotes the nonnegative real valued continuous functions on L. We define the concept of z-ideals in this  semiring and  give a characterization of  its   z-ideals in terms of cozero elements of L. Also, we show that there is a one-one correspondence between  z-ideals and z-congruences on a ring \mathcal{R}(L) and a semiring \mathcal{R}^+(L).  We establish a relationship between z-congruence relation on \mathcal{R}(L) and z-congruence relation on \mathcal{R}^+(L).  A new characterization of P-frames is given via    z-congruences on \mathcal{R}^+(L). Also, we show that there is a bijection between the minimal prime ideals  of \mathcal{R}(L) and  coz-ultrafilter on L.

کلیدواژه ها:

Semiring ، z-congruence ، z-ideal ، coz-filter ، k-ideal