Comprehensive Lecture Notes on Algebraic Structures with One and Two Operations

Foundations of Set Theory and Number Systems

• The concept of a set constitutes a primitive, fundamental mathematical notion that is left formally undefined.

• Standard sets of numbers and their foundational definitions:

  • Natural numbers: $\mathbb{N} = \{1, 2, 3, \dots\}$

  • Extended natural numbers: $\mathbb{N}_0 = \{0, 1, 2, 3, \dots\}$

  • Integers: $\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$

  • Rational numbers: $\mathbb{Q} = \{\frac{p}{q} \mid p \in \mathbb{Z}, q \in \mathbb{N}\}$

  • Real numbers: $\mathbb{R}$

  • Complex numbers: $\mathbb{C} = \{x + i \cdot y \mid x, y \in \mathbb{R}\}$

• Chain of set inclusion:

  • The continuous nested inclusion holds across the fundamental number systems: $\mathbb{N} \subset \mathbb{N}_0 \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$

• Structural classification of algebraic systems:

  • Systems with a single binary operation encompass groupoids, semigroups, monoids, groups, group homomorphisms, and specific geometric/algebraic group models.

  • Systems with two binary operations encompass rings, subrings, division rings (skew fields), and fields.

Monoids and the Uniqueness of Inverses

• Theorem on the equality of one-sided inverses:

  • Theorem: In a monoid $(G, \circ)$, if an element $a \in G$ possesses a left inverse element $a'$ and a right inverse element $a''$, then $a' = a''$.

  • Proof:

  • By the existence of the identity element $e \in G$ and the definition of a left inverse, $a' = a' \circ e$.

  • Substituting the right inverse relation $a \circ a'' = e$ yields $a' = a' \circ (a \circ a'')$.

  • Applying the associative property of the monoid operation $\circ$, this expression rewrites as $a' = (a' \circ a) \circ a''$.

  • Substituting the left inverse relation $a' \circ a = e$ gives $a' = e \circ a''$.

  • By the neutral property of $e$, $e \circ a'' = a''$.

  • Thus, $a' = a''$.

• Theorem on the uniqueness of the inverse element:

  • Theorem: In a monoid $(G, \circ)$, any element $a \in G$ has at most one inverse element.

  • Proof:

  • Assume the contrary: suppose element $a \in G$ possesses two distinct inverse elements $a'$ and $a''$ in the monoid $(G, \circ)$.

  • By the definition of an inverse element, $a' \circ a = a \circ a' = e$ and $a'' \circ a = a \circ a'' = e$.

  • Evaluating $a'$ using the identity element and associativity gives: $a' = a' \circ e = a' \circ (a \circ a'') = (a' \circ a) \circ a'' = e \circ a'' = a''$.

  • This equality contradicts the assumption of distinct inverse elements, proving that the inverse element is unique.

Operational Analysis of Finite Groupoids

• Specification of the finite structure:

  • Consider the set $G = \{1, 2, 3, 4, 5\}$ equipped with a binary operation $\circ$ defined via the following Cayley table:     $\begin{array}{c|ccccc} \circ & 1 & 2 & 3 & 4 & 5 \\ \hline 1 & 1 & 2 & 3 & 4 & 5 \\ 2 & 2 & 2 & 1 & 1 & 4 \\ 3 & 3 & 4 & 1 & 2 & 5 \\ 4 & 4 & 1 & 5 & 1 & 3 \\ 5 & 5 & 1 & 2 & 3 & 3 \end{array}$

• Determination of the neutral element:

  • The neutral (identity) element of the groupoid $(G, \circ)$ is $1$, because $1 \circ x = x$ and $x \circ 1 = x$ for all $x \in G$.

• Evaluation of one-sided inverses for element $2$:

  • Right inverses of $2$: Elements $x \in G$ satisfying $2 \circ x = 1$.

  • $2 \circ 3 = 1$

  • $2 \circ 4 = 1$

  • Thus, the elements $3$ and $4$ are right inverse elements of $2$.

  • Left inverses of $2$: Elements $y \in G$ satisfying $y \circ 2 = 1$.

  • $4 \circ 2 = 1$

  • $5 \circ 2 = 1$

  • Thus, the elements $4$ and $5$ are left inverse elements of $2$.

• Non-associativity of the binary operation:

  • The operation $\circ$ fails to satisfy the associative law, demonstrated by the counterexample:

  • Left-associated composition: $5 \circ (2 \circ 3) = 5 \circ 1 = 5$

  • Right-associated composition: $(5 \circ 2) \circ 3 = 1 \circ 3 = 3$

  • Since $5 \neq 3$, $5 \circ (2 \circ 3) \neq (5 \circ 2) \circ 3$, confirming that the groupoid is not associative.

Geometric and Complex Group Structures

• Rotational symmetry group of the regular triangle:

  • Consider the structure $(\{1, \phi, \psi\}, \circ)$, representing the rotational symmetries of an equilateral triangle, with Cayley table:     $\begin{array}{c|ccc} \circ & 1 & \phi & \psi \\ \hline 1 & 1 & \phi & \psi \\ \phi & \phi & \psi & 1 \\ \psi & \psi & 1 & \phi \end{array}$

  • This algebraic structure is isomorphic to the group $(S, +_3)$, where $S = \{0, 1, 2\}$ and $+_3$ denotes modular addition modulo $3$.

  • Generalization to regular $n$-gons: The set of rotations in the positive (counterclockwise) direction about the circumcenter of a regular $n$-gon by angles $\phi_k = k \cdot \frac{360^\circ}{n}$, where $k \in \{0, 1, \dots, n - 1\}$, forms a cyclic group of order $n$ under composition, isomorphic to $(S_n, +_n)$.

• Multiplicative group of complex cube roots of unity:

  • The set of all solutions to the algebraic equation $z^3 = 1$ in $\mathbb{C}$ consists of:

  • $z_1 = 1$

  • $z_2 = -\frac{1}{2} + i \cdot \frac{\sqrt{3}}{2}$

  • $z_3 = -\frac{1}{2} - i \cdot \frac{\sqrt{3}}{2}$

  • The algebraic structure $(\{z_1, z_2, z_3\}, \cdot)$ under complex multiplication possesses the Cayley table:     $\begin{array}{c|ccc} \cdot & z_1 & z_2 & z_3 \\ \hline z_1 & z_1 & z_2 & z_3 \\ z_2 & z_2 & z_3 & z_1 \\ z_3 & z_3 & z_1 & z_2 \end{array}$

  • This group is isomorphic to the modular group $(S, +_3)$.

  • Generalization to roots of order $n$: The complete set of complex solutions to the equation $z^n = 1$ forms a multiplicative cyclic group of order $n$.

• Symmetry group of a rectangle:

  • Let rectangle $ABCD$ have its axes of symmetry aligned with the coordinate axes $Ox$ and $Oy$.

  • Define the geometric mappings on the vertices of $ABCD$:

  • $X$: Reflection across the coordinate axis $Ox$.

  • $Y$: Reflection across the coordinate axis $Oy$.

  • $O = X \circ Y$: Composition of reflections across $Ox$ and $Oy$, representing a half-turn (central symmetry) with respect to the coordinate origin.

  • $1$: Identity transformation of the rectangle onto itself.

  • The set of mappings $\{X, Y, O, 1\}$ under transformation composition $\circ$ forms a group (the Klein four-group).

Axiomatic Theory of Rings

• Definitions of ring classifications:

  • Commutative ring: A ring $(R, +, \cdot)$ in which the multiplication operation $\cdot$ is commutative.

  • Ring with identity: A ring $(R, +, \cdot)$ in which the multiplication operation $\cdot$ possesses a neutral element. This neutral element is termed the unit element (identity), denoted $1_R$.

  • Example: The algebraic structure $(\mathbb{Z}, +, \cdot)$ is a commutative ring with identity.

• Fundamental operational laws in a ring (Theorem 22):

  • In any arbitrary ring $(R, +, \cdot)$, with additive neutral element $0$:

  1. Multiplicative property of zero: $0 \cdot x = x \cdot 0 = 0$ for all $x \in R$.

  2. Signs in products: $-(x \cdot y) = (-x) \cdot y = x \cdot (-y)$ for all $x, y \in R$.

  3. Product of negatives: $(-x) \cdot (-y) = x \cdot y$ for all $x, y \in R$.

• Proofs of ring properties:

  • Proof of property 1 ($0 \cdot x = 0$):

  • $0 \cdot x = (0 + 0) \cdot x = (0 \cdot x) + (0 \cdot x)$.

  • Because $(R, +)$ is a group, adding the additive inverse of $0 \cdot x$ to both sides yields $0 \cdot x = 0$.

  • By identical reasoning, $x \cdot 0 = x \cdot (0 + 0) = (x \cdot 0) + (x \cdot 0)$, which implies $x \cdot 0 = 0$.

  • Proof of property 2 ($-(x \cdot y) = (-x) \cdot y$):

  • By definition of the additive inverse in $(R, +)$, $(x \cdot y) + [-(x \cdot y)] = 0$.

  • Evaluating the candidate element: $(x \cdot y) + [(-x) \cdot y] = [x + (-x)] \cdot y = 0 \cdot y = 0$.

  • The inverse element of every element in the additive group $(R, +)$ is unique. Therefore, $-(x \cdot y) = (-x) \cdot y$.

  • Symmetrically, $(x \cdot y) + [x \cdot (-y)] = x \cdot [y + (-y)] = x \cdot 0 = 0$, yielding $-(x \cdot y) = x \cdot (-y)$.

  • Proof of property 3 ($(-x) \cdot (-y) = x \cdot y$):

  • Applying property 2 repeatedly: $(-x) \cdot (-y) = -[x \cdot (-y)] = -[-(x \cdot y)]$.

  • Both $-(-(x \cdot y))$ and $x \cdot y$ are additive inverses of the element $-(x \cdot y)$ in the group $(R, +)$.

  • By the uniqueness of the additive inverse, $-(-(x \cdot y)) = x \cdot y$.

  • Hence, $(-x) \cdot (-y) = x \cdot y$.

• Subrings:

  • Let $(R, +, \cdot)$ be a ring. If a non-empty subset $P \subseteq R$ forms a ring under the induced operations $+$ and $\cdot$, then $(P, +, \cdot)$ is called a subring of the ring $(R, +, \cdot)$.

• Ring homomorphisms and isomorphisms:

  • Let $(R, +, \cdot)$ and $(P, \oplus, \odot)$ be rings with multiplicative identities $1_R$ and $1_P$, respectively.

  • A mapping $h: R \to P$ is a ring homomorphism if it satisfies three conditions:

  1. $h(a + b) = h(a) \oplus h(b)$ for all $a, b \in R$

  2. $h(a \cdot b) = h(a) \odot h(b)$ for all $a, b \in R$

  3. $h(1_R) = 1_P$

  • If a ring homomorphism $h$ is bijective, it is termed a ring isomorphism.

Division Rings and Fields

• Axiomatic definition of a division ring (skew field / body):

  • An algebraic structure $(T, +, \cdot)$ equipped with two binary operations is called a division ring if:

  1. $(T, +)$ is an abelian group.

  2. $(T \setminus \{0\}, \cdot)$ is a group, where $0$ is the neutral element of $(T, +)$.

  3. The binary operation $\cdot$ is distributive with respect to the binary operation $+$.

  • Equivalent formulation: A ring with identity in which every element distinct from the additive neutral element $0$ is invertible with respect to multiplication $\cdot$ is a division ring.

• Definition of a field:

  • A division ring $(T, +, \cdot)$ in which the multiplication operation $\cdot$ is commutative is called a field.

• Standard examples of fields:

  • The number systems $(\mathbb{Q}, +, \cdot)$, $(\mathbb{R}, +, \cdot)$, and $(\mathbb{C}, +, \cdot)$ under standard addition and multiplication are fields.

Field Structure on the Set of Real Numbers

• Definition of alternative operations on $\mathbb{R}$:

  • Let operations $\circ$ and $\ast$ on the set $\mathbb{R}$ be defined by:

  • $x \circ y = x + y + 1$

  • $x \ast y = x + y + x \cdot y$

• Additive group properties of $(\mathbb{R}, \circ)$:

  • The structure $(\mathbb{R}, \circ)$ is an abelian group.

  • Neutral element for $\circ$: Determined by $x \circ e_\circ = x$, which expands to $x + e_\circ + 1 = x$, giving $e_\circ = -1$.

  • Inverse element under $\circ$: For any $x \in \mathbb{R}$, the inverse $x'$ satisfies $x \circ x' = -1$, which expands to $x + x' + 1 = -1$, giving $x' = -2 - x$.

• Multiplicative group properties on $\mathbb{R} \setminus \{-1\}$:

  • The structure $(\mathbb{R} \setminus \{-1\}, \ast)$ is an abelian group.

  • Neutral element for $\ast$: Determined by $x \ast e_\ast = x$, which expands to $x + e_\ast + x \cdot e_\ast = x$, or $e_\ast \cdot (1 + x) = 0$. For $x \neq -1$, this requires $e_\ast = 0$.

  • Inverse element under $\ast$: For any $x \in \mathbb{R} \setminus \{-1\}$, the inverse $x''$ satisfies $x \ast x'' = 0$, expanding to $x + x'' + x \cdot x'' = 0$, or $x'' \cdot (1 + x) = -x$, giving $x'' = \frac{-x}{x + 1}$.

• Verification of distributivity of $\ast$ over $\circ$:

  • Because the operation $\ast$ is commutative ($x \ast y = x + y + x \cdot y = y + x + y \cdot x = y \ast x$), verifying left distributivity suffices to establish full distributivity:     $(\forall x, y, z \in \mathbb{R}) \quad x \ast (y \circ z) = (x \ast y) \circ (x \ast z)$

  • Expansion of the left-hand side ($L$):

  • $L = x \ast (y \circ z)$

  • $L = x \ast (y + z + 1)$

  • $L = x + (y + z + 1) + x \cdot (y + z + 1)$

  • $L = x + y + z + 1 + x \cdot y + x \cdot z + x$

  • $L = 1 + 2x + y + z + x \cdot y + x \cdot z$

  • Expansion of the right-hand side ($D$):

  • $D = (x \ast y) \circ (x \ast z)$

  • $D = (x + y + x \cdot y) \circ (x + z + x \cdot z)$

  • Applying the definition $u \circ v = u + v + 1$:

  • $D = (x + y + x \cdot y) + (x + z + x \cdot z) + 1$

  • $D = 1 + 2x + y + z + x \cdot y + x \cdot z$

  • Conclusion: Because $L = D$ for all $x, y, z \in \mathbb{R}$, distributivity holds. Consequently, the algebraic structure $(\mathbb{R}, \circ, \ast)$ is a field.