On semigroup algebras with rational exponents

In this paper, a semigroup algebra consisting of polynomial expressions with coefficients in field F and exponents an additive submonoid M Q≥0 is called Puiseux denoted by F[M]. Here we study the atomic structure algebras. To begin with, answer isomorphism problem for class algebras, that is, show if two algebras F[M1] F[M2] are isomorphic, then monoids M1 M2 must be isomorphic. Then construct three classes satisfying following well-known properties: ACCP property, bounded factorization finite property. We there extremal systems sets lengths, which allows us to prove cannot determined (up isomorphism) their arithmetic lengths. Finally, give full description seminormal closure, root complete integral closure algebra, use provide antimatter (i.e., containing no irreducibles).