Association schemes and hypergroups

Abstract

Abstract. In this paper, we investigate hypergroups which arise from association schemes in a canonical way; this class of hypergroups is called realizable. We first study basic algebraic properties of realizable hypergroups. Then we prove that two interesting classes of hypergroups (partition hypergroups and linearly ordered hypergroups) are realizable. Along the way, we prove that a certain class of projective geometries is equipped with a canonical association scheme structure which allows us to link three objects; association schemes, hypergroups, and projective geometries (see, §1.2 for details).

New Aspects in the Theory of Complete Hypergroups

“…Starting in 1934, when F. Marty introduced the hypergroup as a new algebraic structure extending the classical one of the group, and satisfying the same properties, i.e., the associativity and reproductivity, the theory of hypercompositional structures (also called hyperstructure theory) has experienced a rapid growth, succeeding to impose itself as a branch of Abstract Algebra. Currently, this theory offers a strong background for studies in algebraic geometry [1], number theory [2], automata theory [3], graph theory [4], matroids theory [5], and association schemes [6], to list just some of the research fields where hypercompositional structures are deeply involved. These structures are non-empty sets endowed with at least one hyperoperation, i.e., a multivalued operation resulting in a subset of the underlying set, usually denoted as • : H × H −→ P * (H), where (P) * (H) denotes the set of non-empty subsets of H. A non-empty set H equipped with a hyperoperation that satisfies: (i) the associativity: (x • y) • z = x • (y • z) for any x, y, z ∈ H, where (x • y) • z = u∈x•y u • z and x • (y • z) = v∈y•z x • v, and (ii) the reproductivity:…”

From HX-Groups to HX-Polygroups

“…Its commutative version, i.e., the canonical hypergroup, dates back to the beginning of 1970s, when Mittas [25] studied it as an independent structure in the framework of valuation theory, and not just as the additive structure of a hyperfield. In fact, this was the way that canonical hypergroups appeared in the first studies of Krasner [26] and have continued to be investigated as the additive structure of the Krasner hyperfields and the hypercompositional structure with the most applications in different areas, e.g., valuation theory [27][28][29], algebraic geometry [30], number theory, affine algebraic group schemes [31], matroids theory [32], tropical geometry [33], and hypermodules [34]. The state of the art in hyperfield theory was included in an article recently published by Ch.…”

Proto-exact categories of modules over semirings and hyperrings

“…The main goal of the current paper is to enlarge the catalogue of non-additive proto-exact categories by showing these include the categories of modules over semirings as well as hyperrings. Modules over an idempotent semiring are closely related to matroid theory [GG18] and modules over a hyperring have an interesting connection to finite incidence geometries [CC10a], [Jun18b] and matroids [BB19]. We also examine the category of algebraic lattices in relation to finite modules over B, and discuss how the proto-exact structure of algebraic lattices is related to the proto-exact structure of the category of matroids in [EJS20] via geometric lattices.…”