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Algebraic Structures on the Set of all Binary Operations over a Fixed Set

Owusu-Mensah, Isaac
http://orcid.org/0000-0003-3488-4361
http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1584490788584639

Abstract Details

2020, Doctor of Philosophy (PhD), Ohio University, Mathematics (Arts and Sciences).
The word magma is often used to designate a pair of the form (S,*) where * is a binary operation on the set S. We use the notation Μ(S)(the magma of S to denote the set of all binary operations on the set S (i.e. all magmas with underlying set S.) Our work on this set was motivated initially by an intention to better understand the distributivity relation among operations over a fixed set; however, our research has yielded structural and combinatoric questions that are interesting in their own right. Given a set S, its (left, right, two-sided) hierarchy graph is the directed graph that has Μ(S)(as its set of vertices and such that there is an edge from an operation ⋆ to another one ⋄ if ⋆ distributes over ⋄(on the left, right, or both sides.) The graph-theoretic setting allows us to describe easily various interesting algebraic scenarios. We study several combinatorial properties of hierarchy graphs including the length of a longest cycle-free path in the hierarchy graph and the largest possible cardinality of a complete subgraph. Of particular interest in this study are the collections of operations defined next: given ⋆ ∈ Μ(S), the outset of ⋆, is the set out(⋆)={ο∈M(S)┤| ⋆ distributes over ο }. We define an operation ⊲ that makes Μ(S) a monoid in such a way that each outset out(⋆) is a submonoid of (Μ(S),⊲).This endowment gives us a possibility to explore the properties of an element ⋆ ∈ Μ(S), in terms of the structure of its outset monoid (out(⋆),⊲). Other connections between the ⊲ operation and the hierarchy graph are studied. Among the properties of the ⊲ operation that are considered are various submonoids, one-sided, and two-sided ideals. A complete description of its group of units is given and its group of automorphisms is studied. In addition, multiple choices for an additive binary operations + on (M(S) are given so that ⊲ may be considered the multiplicative operation at the bottom of a nearring (M(S),+,⊲) . Our study of magmas that enjoy special behaviors in the context of the magma monoid led us to the discussion of certain operations induced by graphs; at least two of them are of special note: the one-value magmas and the two-value magmas. Our study was carried through the traditional strategies of mathematical inquiry but aided at times with experimentation with the symbolic computation software package MAGMA.
Sergio Lopez-Permouth, Prof (Advisor)
Alexei Davydov, Prof (Committee Member)
Bischoff Marcel , Prof (Committee Member)
Sergio Ulloa , Prof (Committee Member)
91 p.
Mathematics
Algebraic Structures; Binary Operations; Monoid; Distributivity; Nearrings; Graph Magma

Recommended Citations

Citations

  • Owusu-Mensah, I. (2020). Algebraic Structures on the Set of all Binary Operations over a Fixed Set [Doctoral dissertation, Ohio University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1584490788584639

    APA Style (7th edition)

  • Owusu-Mensah, Isaac. Algebraic Structures on the Set of all Binary Operations over a Fixed Set. 2020. Ohio University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1584490788584639.

    MLA Style (8th edition)

  • Owusu-Mensah, Isaac. "Algebraic Structures on the Set of all Binary Operations over a Fixed Set." Doctoral dissertation, Ohio University, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=ohiou1584490788584639

    Chicago Manual of Style (17th edition)

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