This thesis will contain a detailed overview of relations, quotients, normality, loops, quasigroups, and related theorems and varieties. Nonassociative algebra is a relatively new area of mathematics, it came about in the past hundred years, and has started making progress in the past 60 years. In nonassociative algebra, varieties do not necessarily satisfy associativity. Several interesting problems with relations, quotients, and normality arise from the setting of nonassociative algebra. In the language of equivalence relations, quotients, and subsets what are the conditions of normality, or existence of a subalgebra in quasigroups and loops? A quasigroup, Q, is defined to be algebra (Q, ·,\\,/) that takes two elements from the quasigroup, under one of the operations ·,\\,/, and maps the combination to another element of the quasigroup. The three quasigroup binary operations must satisfy axioms that guarantee left and right inverses; additionally the inverses are unique. A loop, typically denoted (L, ·), is a quasigroup with a two-sided identity element e. For e to be a two sided identity element, e · x = x · e = x for any x in L. In order for a relation to be an equivalence relation it must be reflexive, symmetric, and transitive.
","abstract_html":"<p>This thesis will contain a detailed overview of relations, quotients, normality, loops, quasigroups, and related theorems and varieties. Nonassociative algebra is a relatively new area of mathematics, it came about in the past hundred years, and has started making progress in the past 60 years. In nonassociative algebra, varieties do not necessarily satisfy associativity. Several interesting problems with relations, quotients, and normality arise from the setting of nonassociative algebra. In the language of equivalence relations, quotients, and subsets what are the conditions of normality, or existence of a subalgebra in quasigroups and loops? A quasigroup, Q, is defined to be algebra (Q, ·,\\,/) that takes two elements from the quasigroup, under one of the operations ·,\\,/, and maps the combination to another element of the quasigroup. The three quasigroup binary operations must satisfy axioms that guarantee left and right inverses; additionally the inverses are unique. A loop, typically denoted (L, ·), is a quasigroup with a two-sided identity element e. For e to be a two sided identity element, e · x = x · e = x for any x in L. In order for a relation to be an equivalence relation it must be reflexive, symmetric, and transitive.</p>","abstract_has_math":false,"creators":["Mulholland, Matthew L"],"institution":null,"degree_name":"Master of Science","degree_level":"Thesis","degree_discipline":null,"degree_department":null,"school":null,"contributors":["Dr. Daniel Rowe"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2025,"date_issued":"2025-05-01T07:00:00Z","date_published":"2025-05-01T07:00:00Z","updated_at":"2026-07-24T03:24:36Z","subjects":["Quotients","Relations","Normality","Non-Associative Algebra","Quasigroups","Loops","Algebra"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://commons.nmu.edu/theses/870","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Dr. Daniel Rowe"]},{"key":"dc:creator","label":"Author","values":["Mulholland, Matthew L"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2025-04-04T07:00:00Z"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Quotients","Relations","Normality","Non-Associative Algebra","Quasigroups","Loops","Algebra"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://commons.nmu.edu/theses/870"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This thesis will contain a detailed overview of relations, quotients, normality, loops, quasigroups, and related theorems and varieties. Nonassociative algebra is a relatively new area of mathematics, it came about in the past hundred years, and has started making progress in the past 60 years. In nonassociative algebra, varieties do not necessarily satisfy associativity. Several interesting problems with relations, quotients, and normality arise from the setting of nonassociative algebra. In the language of equivalence relations, quotients, and subsets what are the conditions of normality, or existence of a subalgebra in quasigroups and loops? A quasigroup, Q, is defined to be algebra (Q, ·,\\,/) that takes two elements from the quasigroup, under one of the operations ·,\\,/, and maps the combination to another element of the quasigroup. The three quasigroup binary operations must satisfy axioms that guarantee left and right inverses; additionally the inverses are unique. A loop, typically denoted (L, ·), is a quasigroup with a two-sided identity element e. For e to be a two sided identity element, e · x = x · e = x for any x in L. In order for a relation to be an equivalence relation it must be reflexive, symmetric, and transitive.
"]},{"key":"dc:title","label":"Title","values":["QUOTIENTS, EQUIVALENCE RELATIONS, AND NORMALITY IN NON-ASSOCIATIVE ALGEBRA WITH REGARDS TO LOOPS AND QUASIGROUPS"]}]}],"canonical_facts":{"dc:contributor":["Dr. Daniel Rowe"],"dc:creator":["Mulholland, Matthew L"],"dc:date.available":["2025-04-04T07:00:00Z"],"dc:description.abstract":["This thesis will contain a detailed overview of relations, quotients, normality, loops, quasigroups, and related theorems and varieties. Nonassociative algebra is a relatively new area of mathematics, it came about in the past hundred years, and has started making progress in the past 60 years. In nonassociative algebra, varieties do not necessarily satisfy associativity. Several interesting problems with relations, quotients, and normality arise from the setting of nonassociative algebra. In the language of equivalence relations, quotients, and subsets what are the conditions of normality, or existence of a subalgebra in quasigroups and loops? A quasigroup, Q, is defined to be algebra (Q, ·,\\,/) that takes two elements from the quasigroup, under one of the operations ·,\\,/, and maps the combination to another element of the quasigroup. The three quasigroup binary operations must satisfy axioms that guarantee left and right inverses; additionally the inverses are unique. A loop, typically denoted (L, ·), is a quasigroup with a two-sided identity element e. For e to be a two sided identity element, e · x = x · e = x for any x in L. In order for a relation to be an equivalence relation it must be reflexive, symmetric, and transitive.
"],"dc:identifier":["https://commons.nmu.edu/theses/870"],"dc:subject":["Quotients","Relations","Normality","Non-Associative Algebra","Quasigroups","Loops","Algebra"],"dc:title":["QUOTIENTS, EQUIVALENCE RELATIONS, AND NORMALITY IN NON-ASSOCIATIVE ALGEBRA WITH REGARDS TO LOOPS AND QUASIGROUPS"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["Master of Science"]},"updated_at":"2026-07-24T03:24:36Z"}