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    "Abstract Algebra Types\n",
    "=============="
   ]
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   "source": [
    "## Group\n",
    "\n",
    "Concrete Implementation: `klefki.types.algebra.abstract.Group`\n",
    "\n",
    "A set $\\mathbb{G}={a, b, c, ...}$ is called a group, if tehere exists a group addition $(+)$ connecting the elements in $(\\mathbb{G}, +)$ in the following way:\n",
    "\n",
    "(1) $a, b \\in \\mathbb{G}:\\ c=a+b \\in \\mathbb{G}$ (closure)\n",
    "\n",
    "(2) $a, b, c \\in \\mathbb{G}: (a+b)c=a(b+c)$ (associativity)\n",
    "\n",
    "(3) $\\exists e \\in \\mathbb{G}: a+e=e, \\forall a \\in \\mathbb{G}$ (identity / neutral element)\n",
    "\n",
    "(4) $\\forall a \\in \\mathbb{G}, \\exists b \\in \\mathbb{G}: a+b=e, i.e., b\\equiv -a$ (inverse)\n",
    "\n",
    "if a group obey axiom (1,2), it is a SemiGroup;\n",
    "\n",
    "if a group obey axiom (1,2,3), it is a monadid;\n",
    "\n",
    "if a group obey axiom (1,2,3,4) and the axiom of commutatativity($a+b=b+a$), it is a Abelian Group\n",
    "\n"
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   "source": [
    "## Field\n",
    "\n",
    "Concrete Implementation: `klefki.types.algebra.abstract.Field`\n",
    "\n",
    "A field is any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.\n",
    "\n",
    "Field Axioms\\cite{FieldAxioms} are generally written in additive and multiplication pairs:\n",
    "\n",
    "(1) $(a+b)+c=a+(b+c)$; $(a b) c = a(b c)$ (associativity)\n",
    "\n",
    "(2) $a + b = b + a$; $ a b = b a$ (Commutativity)\n",
    "\n",
    "(3) $a(b+c) = ab + ac$; $(a+b)c=ac+bc$ (distributivity)\n",
    "\n",
    "(4) $a + 0=a=0+1$; $(a.1=a=1.a)$ (identity)\n",
    "\n",
    "(5) $a+(-a)=0=(-a)+a$; $aa^{-1}=1=a^{-1}a if a \\neq 0$ (inverses)\n"
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    "## The Finite Field\n",
    "\n",
    "Concrete Implementation: `klefki.types.algebra.fields.FiniteField`\n",
    "\n",
    "A finite field is, A set with a finite number of elements. An example of inite field is the set of integers modulo $p$, where $p$ is a prime number, which can be generally note as $\\mathbb{Z}/p$, $GF(p)$ or $\\mathbb{F}_p$.\n",
    "\n"
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