latexit.md

LaTeX Snippets

This file contains the LaTeX snippets which the OmniGraffle document includes via LaTeXiT. Unless noted otherwise, the font size in LaTeXiT is 9 pt.

Algebra

\begin{tabular}{ll}
\textsc{Algebra} & $\langle S,\Phi \rangle$\\
\textsc{Carrier Set} & $S$\\
\textsc{Operator Set} & $\Phi = \{f_1,\ldots,f_n\}$\\
\textsc{Operator} & $f_i: S^m \to S$\\
\textsc{Arity} & $m \in \mathbb{N}_0$\\
%\textsc{Arity} & $\mathbb{A}(f_i) = m \in \mathbb{N}_0$\\
%\textsc{Signature} & $\{\mathbb{A}(f_i) \mid f_i \in \Phi\}$
\end{tabular}

Font size: 12pt

Magma

&\langle S,\{\circ\}\rangle\\
&\circ:S\times S \to S\\
%&\mathbb{A}(\circ) = 2

Associativity

\centering
\textsc{Associativity}\\
$(x \circ y) \circ z = x \circ (y \circ z)$

Identity

\centering
\textsc{Identity}\\
$1 \circ x = x \circ 1 = x$

Inverse

\centering
\textsc{inverse}\\
${-x} \circ x = x \circ {-x} = 1$

Commutativity

\centering
\textsc{Commutativity}\\
$x \circ y = y \circ x$

Semiring

&\langle S, \{\oplus,\odot\} \rangle\\
&\oplus: S \times S \to S\\
&\odot: S \times S \to S\\
%&\mathbb{A}(\oplus) = 2\\
%&\mathbb{A}(\odot) = 2

1.\; & \textsc{Commutative Monoid}\;\langle S, \oplus\rangle\\ 
     &\textsc{Additive Identity}\;0\in S\\ 
2.\; & \textsc{Monoid}\;\langle S, \odot\rangle\\
     &\textsc{Multiplicative Identity}\;1\in S\\ 
3.\; & \textsc{Distributivity}\\
& x \odot (y \oplus z) = (x \odot y) \oplus (x \odot z)\\
& (y \oplus z) \odot x = (y \odot x) \oplus (z \odot x) \\

Semiring -> Ring

\centering
\textsc{Commutative Group}\\
$\langle S, \oplus \rangle$

Ring -> Commutative Ring

\centering
\textsc{Commutative Monoid}\\
$\langle S, \odot \rangle$

Commutative Ring -> Integral Domain

\centering
\textsc{No Zero Divisors}\\
$\forall x,y \in S\setminus\{0\}: xy \ne 0$

Ring -> Field

\centering
\textsc{Commutative Group}\\
$\langle S, \odot \rangle$