Decompositions in Semirings

Abstract

We prove that each element of a complete atomic \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$ l $\\end{document}-semiring has a canonical decomposition.We also find some sufficient conditions for the decomposition to be uniquethat are expressed by first-order sentences.As a corollary, we obtain a theorem of Avgustinovich–Frid which claims thateach factorial language has the unique canonical decomposition.