Geodesic
Additive Divisibility in Compact Topological Semirings
Canadian journal of mathematics, Tome 26 (1974) no. 3, pp. 593-599

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A topological semiring (S, + , ·) is a nonempty Hausdorff space S on which are defined continuous and associative operations, termed addition (+) and multiplication (·), such that the multiplication distributes over addition from left and right. The additive semigroup (S, +) need not be commutative.We prove that the set A of additively divisible elements of a compact semiring S is a two-sided multiplicative ideal, containing the set E[+] of additive idempotents, with the property that (A.S) ∪ (S.A) ⊂ E[+]. 
Karvellas, P. H. Additive Divisibility in Compact Topological Semirings. Canadian journal of mathematics, Tome 26 (1974) no. 3, pp. 593-599. doi: 10.4153/CJM-1974-056-7
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