Mathematics - Discrete Mathematics - Group-like Structures (part 2)

[Image 1]

Introduction

Hey it's a me again @drifter1!

Today we continue with Mathematics, and more specifically the branch of "Discrete Mathematics", in order to get into Group-like Structures. This is part 2, you can find part 1 here.

So, without further ado, let's get straight into it!

---

Magma

Easing the requirements of the definition of groups to the point that only the closure property has to be satisfied, yields an algebraic structure known as a Magma. A magma is thus a non-empty set together with a binary operation that satisfies closure (basically any function).

A magma clearly has way to few properties (only one), so let's add one additional property to come up with other structures...

Associative Magma (or Semigroup)

A magma that also satisfies associativity is basically a semigroup.

Unital Magma

A magma that satisfies identity is called an Unital magma.

Quasigroup

A magma which satisfies invertibility (or division), which basically means that division is always possible between any pair of elements is called a Quasigroup. A quasigroup doesn't have to be associative, nor does it have to include an identity element.

Loop

Adding an identity element to a Quasigroup yields a structure known as a Loop, which satisfies closure, identity and division.

---

Groupoid

A groupoid is commonly defined as a structure that doesn't satisfy closure but satisfies all other properties of a group (associativity, identity and invertibility). A groupoid basically easies the requirement of the binary operation, so that it doesn't have to be defined for all elements in the set.

Semigroupoid

A semigroupoid is a structure that satisfies only associativity.

Small category

A small category is a structure that satisfies associativity and has an identity element.

---

RESOURCES:

References

1. https://www.javatpoint.com/discrete-mathematics-tutorial
2. https://mathworld.wolfram.com/Group.html
3. https://en.wikipedia.org/wiki/Group_(mathematics)

Images

1. https://commons.wikimedia.org/wiki/File:Semigroup_associative.svg

Mathematical equations used in this article, have been generated using quicklatex.

Block diagrams and other visualizations were made using draw.io.

---

Previous articles of the series

• Introduction → Discrete Mathematics, Why Discrete Math, Series Outline
• Sets → Set Theory, Sets (Representation, Common Notations, Cardinality, Types)
• Set Operations → Venn Diagrams, Set Operations, Properties and Laws
• Sets and Relations → Cartesian Product of Sets, Relation and Function Terminology (Domain, Co-Domain and Range, Types and Properties)
• Relation Closures → Relation Closures (Reflexive, Symmetric, Transitive), Full-On Example
• Equivalence Relations → Equivalence Relations (Properties, Equivalent Elements, Equivalence Classes, Partitions)
• Partial Order Relations and Sets → Partial Order Relations, POSET (Elements, Max-Min, Upper-Lower Bounds), Hasse Diagrams, Total Order Relations, Lattices
• Combinatorial Principles → Combinatorics, Basic Counting Principles (Additive, Multiplicative), Inclusion-Exclusion Principle (PIE)
• Combinations and Permutations → Factorial, Binomial Coefficient, Combination and Permutation (with / out repetition)
• Combinatorics Topics → Pigeonhole Principle, Pascal's Triangle and Binomial Theorem, Counting Derangements
• Propositions and Connectives → Propositional Logic, Propositions, Connectives (∧, ∨, →, ↔ and ¬)
• Implication and Equivalence Statements → Truth Tables, Implication, Equivalence, Propositional Algebra
• Proof Strategies (part 1) → Proofs, Direct Proof, Proof by Contrapositive, Proof by Contradiction
• Proof Strategies (part 2) → Proof by Cases, Proof by Counter-Example, Mathematical Induction
• Sequences and Recurrence Relations → Sequences (Terms, Definition, Arithmetic, Geometric), Recurrence Relations
• Probability → Probability Theory, Probability, Theorems, Example
• Conditional Probability → Conditional Probability, Law of Total Probability, Bayes' Theorem, Full-On Example
• Graphs → Graph Theory, Graphs (Vertices, Types, Handshake Lemma)
• Graphs 2 → Graph Representation (Adjacency Matrix and Lists), Graph Types and Properties (Isomorphic, Subgraphs, Bipartite, Regular, Planar)
• Paths and Circuits → Paths, Circuits, Euler, Hamilton
• Trees → Trees (Rooted, General and Binary), Tree Traversal, Spanning Trees
• Common Graph Problems → Shortest Path Problem, Graph Connectivity, Travelling Salesman Problem, Minimum Spanning Tree, Maximum Network Flow, Graph Coloring
• Binary Operations → Binary Operations (n-ary, Table Representation), Properties
• Groups → Groups (Properties, Theorems, Finite and Infinite, Abelian, Cyclic, Product, Homo-, Iso- and Auto-morphism)
• Group-like Structures (part 1) → Subgroups, Semigroups, Monoids

---

Final words | Next up

And this is actually it for today's post!

Next time we will cover Rings...

See ya!

Keep on drifting!

Posted with STEMGeeks