On Posner s second theorem in additively inverse semirings

Additively Inverse Semirings

In this paper we show that in a regular additively inverse semiring (S,+, ·) with 1 satisfying the conditions (A) a(a + a′) = a + a′; (B) a(b + b′) = (b + b′)a and (C) a + a(b + b′) = a, for all a, b ∈ S, the sum of two principal left ideals is again a principal left ideal. Also, we decompose S as a direct sum of two mutually inverse ideals.

Kleene ’ s Amazing Second Recursion Theorem Extended

Kleene states the theorem with V = N, relative to specific φ, S n , supplied by his Enumeration Theorem, m = 0 (no parameters ~y) and n ≥ 1, i.e., not allowing nullary partial functions. And most of the time, this is all we need; but there are a few important applications where choosing “the right” φ, S n , restricting the values to a proper V ( N or allowing m > 0 or n = 0 simplifies the proof...

The Cayley-Hamilton Theorem for Noncommutative Semirings

The Cayley-Hamilton theorem (CHT) is a classic result in linear algebra over fields which states that a matrix satisfies its own characteristic polynomial. CHT has been extended from fields to commutative semirings by Rutherford in 1964. However, to the best of our knowledge, no result is known for noncommutative semirings. This is a serious limitation, as the class of regular languages, with f...

A Fundamental Theorem of Homomorphisms for Semirings

1. Introduction. When studying ideal theory in semirings, it is natural to consider the quotient structure of a semiring modulo an ideal. If 7 is an ideal in a semiring R, the collection {x+l}xeit oí sets x + I={x+i\iEl} need not be a partition of R. Faced with this problem, [2] used equivalence relations to determine cosets relative to an ideal. La Torre successfully established analogues of s...

On the Inverse Function Theorem