The main purpose of this article is to initiate a systematic study of Semihypergroups, first introduced by Dunkl (Am Math Soc 179:331–348, 1973 ), Jewett (Adv Math 18(1):1–101, 1975 ) and Spector (Apercu de la theorie des hypergroups, (French) Analyse harmonique sur les groupes de Lie (Sém. Nancy–Strasbourg, 1973–75), Springer, New York, 1975) independently around 1972. We introduce and study several natural algebraic and analytic structures on semihypergroups, which are well-known in the case of topological groups and semigroups. In particular, we first study almost periodic and weakly almost periodic function spaces (basic properties, their relation to the compactness of the underlying space, introversion and Arens product on their duals among others). We then introduce homomorphisms and ideals, and thereby examine their behaviour (basic properties, structure of the kernel and relation of amenability to minimal ideals) in order to gain insight into the structure of a Semihypergroup itself. In the process, we further investigate where and why this theory deviates from the classical theory of semigroups.

中文翻译：

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半超群分析：函数空间、同态和理想

本文的主要目的是启动对半超群的系统研究，首先由 Dunkl (Am Math Soc 179:331–348, 1973 )、Jewett (Adv Math 18(1):1–101, 1975) 和 Spector ( Apercu de la theorie des hypergroups，（法语）Analyzeharmonique sur les groupes de Lie (Sém. Nancy–Strasbourg, 1973–75), Springer, New York, 1975) 于 1972 年左右独立。我们介绍并研究了几种自然代数和解析结构在半超群上，这在拓扑群和半群的情况下是众所周知的。特别是，我们首先研究了几乎周期性和弱几乎周期性的函数空间（基本属性，它们与基础空间的紧凑性的关系，内向性和对偶上的阿伦斯乘积等）。然后我们引入同态和理想，从而检查它们的行为（基本属性、内核结构以及与最小理想的适应性关系），以便深入了解半超群本身的结构。在这个过程中，我们进一步研究了这个理论在何处以及为什么偏离了经典的半群理论。