On the growth of delta-subharmonic functions on an unbounded semiring

This paper considers spaces of delta-subharmonic functions on an open unbounded semiring. We introduce a definition of a delta-subharmonic function of finite $\gamma$-growth on an unbounded semiring in which no restrictions are imposed on the growth function. We obtain criteria for a delta-subharmonic function to belong to this space, which are formulated in terms of the Fourier coefficients of this function. The criteria extend to the semiring the results obtained earlier in a joint work by L.A. Rubel and B.A. Taylor for meromorphic functions on the complex plane and the results of K.G. Malyutin for delta-subharmonic functions on the half-plane.