Abstract
A topological semiring is a triplet (S, +, •) where S is a Hausdorff topological space, “+” and “•” are jointly continuous associative binary operations on S and “•” distributes across “+” on both sides. Recent work by J. Selden [16], K. R. Pearson [13], and Paul H. Karvellas [7] has provided information about and, in some cases, complete characterizations of (S, +, •) when (S, +) or (S, •) are specified. Herein, we consider the case in which (S, •) is the one-point compactification of a closed proper cone in En with vector addition extended so that the point at infinity is a zero. Further, if (S, +) is assumed to be a semilattice, we give a complete characterization of (S, +, •).
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References
Anzai, H., On compact topological rings, Proc. Japan Acad. 19 (1943), pp. 613–615.
Beidleman, J. C. and R. H. Cox, Topological Nearrings, Arch. Math. 18 (1967) pp. 485–492.
Brown, D. R. and M. Friedberg, Representation theorems for uniquely-divisible semigroups, Duke Math. J. Vol. 35, No. 2 (1968), pp. 341–352.
Clifford, A. H. and G. B. Preston, “The Algebraic Theory of Semigroups”, Vol. I, Math. Surveys, 7, Am. Math. Soc. (1961).
Hofmann, K. H. and P. S. Mostert, “Elements of Compact Semigroups”, C. E. Merrill, Inc., Columbus, Ohio (1966).
Kaplansky, I., Topological rings, Amer. J. Math. 69, (1947), pp. 153–183.
Karvellas, P. H., “Algebraic and Topological Semirings”, Dissertation, University of Houston (1972).
——, Additive divisibility in compact topological semirings, Can. J. Math. Vol. XXVI, No. 3 (1974), pp. 593–599.
Keimel, Klaus, Eine Exponentialfunktion für kompakte abelsche Halbgruppen, Math. Z., Vol. 96 (1967), pp. 7–25.
——, Congruence relations on cone semigroups, Semigroup Forum, Vol. 3, No. 2 (1971), pp. 130–147.
Lawson, J. D., Joint continuity in semitopological semigroups, Illinois J. Math. 18 (1974) pp. 275–285.
——, and B. L. Madison, On congruences and cones, Math. Z., 120 (1971) pp. 18–24.
Pearson, K. R., Interval semigroups on R1 with ordinary multiplication, J. Aust. Math. Soc., 6 (1966) pp. 273–288.
——, Certain topological semirings in R1, J. Aust. Math. Soc., 8 (1968), pp. 171–182.
——, The three kernels of a compact semiring, J. Aust. Math. Soc. 10 (1969) pp. 229–319.
Selden, John, “Theorems on Topological Semigroups and Semirings”, Dissertation, University of Georgia (1963).
——, A note on compact semirings, Proc. Am. Math. Soc. 15 (1964), pp. 882–886.
Vulikh, B. Z., “Introduction of the Theory of Partially Ordered Spaces”, Wolters-Noordhoff, Groningen, (1967).
Wallace, A. D., On the structure of topological semigroups, Bull. Am. Math. Soc., 61 (1955), pp. 95–112.
Yudin, A. I., Solution of two problems of the theory of partially ordered spaces, DAN SSR 23 (1939), pp. 418–422 (Russian).
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TO ALEXANDER DONIPHAN WALLACE on his 69th birthday on the 21st of August, 1974.
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Friedberg, M. Semirings on cones. Semigroup Forum 10, 329–350 (1975). https://doi.org/10.1007/BF02194901
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DOI: https://doi.org/10.1007/BF02194901