On completeness of sheaf representations of semirings

The aim of the paper is to study the completeness conditions for sheaf representation of semirings. The proofs extensively use the open congruence sets, the properties of which are investigated in the first section. Some applications of the developed constructions are shown: namely, the Lambek representation for stricly harmonic and stricly semiprime semirings as well as Pierce representations of arbitrary semirings were proved to be isomorphic.