Mathematical Structures: Basic logic algebras

# Basic logic algebras

Difference (from prior major revision) (no other diffs)

Changed: 1,143c1,139
 %%run pdflatex % \documentclass[12pt]{amsart} \usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref} \parindent=0pt \parskip=5pt \addtolength{\oddsidemargin}{-.5in} \addtolength{\evensidemargin}{-.5in} \addtolength{\textwidth}{1in} \theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem*{morphisms}{Morphisms} \newtheorem*{basic_results}{Basic Results} \newtheorem*{examples}{Examples} \newtheorem{example}{} \newtheorem*{properties}{Properties} \newtheorem*{finite_members}{Finite Members} \newtheorem*{subclasses}{Subclasses} \newtheorem*{superclasses}{Superclasses} \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}} \pagestyle{myheadings}\thispagestyle{myheadings} \markboth{\today}{math.chapman.edu/structures} \begin{document} \textbf{\Large Basic logic algebras} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Basic_logic_algebras}{edit} \abbreviation{BLA} \begin{definition} A \emph{basic logic algebra} or \emph{BL-algebra} is a structure $\mathbf{A}=\left\langle A,\vee ,0,\wedge ,1,\cdot ,\rightarrow \right\rangle$ such that $\left\langle A,\vee ,0,\wedge ,1\right\rangle$ is a \href{Bounded_lattices.pdf}{Bounded lattices} $\left\langle A,\cdot ,1\right\rangle$ is a \href{Commutative_monoids.pdf}{Commutative monoids} $\rightarrow$ gives the residual of $\cdot$: $x\cdot y\leq z\Longleftrightarrow y\leq x\rightarrow z$ linearity: $\left( x\rightarrow y\right) \vee \left( y\rightarrow x\right) =1$ BL: $x\cdot(x\rightarrow y)=x\wedge y$ Remark: The BL identity implies that the lattice is distributive. \end{definition} \begin{definition} A \emph{basic logic algebra} is a \href{FLe-algebras.pdf}{FLe-algebras} $\mathbf{A}=\left\langle A,\vee ,0,\wedge ,1,\cdot ,\rightarrow \right\rangle$ such that linearity: $\left( x\rightarrow y\right) \vee \left( y\rightarrow x\right) =1$ BL: $x\cdot (x\rightarrow y)=x\wedge y$ Remark: The BL identity implies that the identity element $1$ is the top of the lattice. \end{definition} \begin{morphisms} Let $\mathbf{A}$ and $\mathbf{B}$ be basic logic algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x\vee y)=h(x)\vee h(y)$, $h(1)=1$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$ \end{morphisms} \begin{basic_results} \end{basic_results} \begin{examples} \begin{example} \end{example} \end{examples} \begin{table}[h] \begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description}) \begin{tabular}{|ll|}\hline Classtype & variety\\\hline Equational theory & decidable\\\hline Quasiequational theory & \\\hline First-order theory & \\\hline Locally finite & no\\\hline Residual size & unbounded\\\hline Congruence distributive & yes\\\hline Congruence modular & yes\\\hline Congruence n-permutable & yes, $n=2$\\\hline Congruence e-regular & yes, $e=1$\\\hline Congruence uniform & no\\\hline Congruence extension property & yes\\\hline Definable principal congruences & \\\hline Equationally def. pr. cong. & no\\\hline Amalgamation property & \\\hline Strong amalgamation property & \\\hline Epimorphisms are surjective & \\\hline \end{tabular} \end{properties} \end{table} \begin{finite_members} $f(n)=$ number of members of size $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \begin{subclasses}\ \href{MV-algebras.pdf}{MV-algebras} \href{Heyting_algebras.pdf}{Heyting algebras} \end{subclasses} \begin{superclasses}\ \href{Generalized_basic_logic_algebras.pdf}{Generalized basic logic algebras} \href{FLew-algebras.pdf}{FLew-algebras} \end{superclasses} \begin{thebibliography}{10} \bibitem{Ln19xx} \end{thebibliography} \end{document} %
 %%run pdflatex % \documentclass[12pt]{amsart} \usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref} \parindent=0pt \parskip=5pt \addtolength{\oddsidemargin}{-.5in} \addtolength{\evensidemargin}{-.5in} \addtolength{\textwidth}{1in} \theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem*{morphisms}{Morphisms} \newtheorem*{basic_results}{Basic Results} \newtheorem*{examples}{Examples} \newtheorem{example}{} \newtheorem*{properties}{Properties} \newtheorem*{finite_members}{Finite Members} \newtheorem*{subclasses}{Subclasses} \newtheorem*{superclasses}{Superclasses} \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}} \pagestyle{myheadings}\thispagestyle{myheadings} \markboth{\today}{math.chapman.edu/structures} \begin{document} \textbf{\Large Basic logic algebras} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Basic_logic_algebras}{edit} \abbreviation{BLA} \begin{definition} A \emph{basic logic algebra} or \emph{BL-algebra} is a structure $\mathbf{A}=\left\langle A,\vee ,0,\wedge ,1,\cdot ,\rightarrow \right\rangle$ such that $\left\langle A,\vee ,0,\wedge ,1\right\rangle$ is a \href{Bounded_lattices.pdf}{bounded lattice} $\left\langle A,\cdot ,1\right\rangle$ is a \href{Commutative_monoids.pdf}{commutative monoid} $\rightarrow$ gives the residual of $\cdot$: $x\cdot y\leq z\Longleftrightarrow y\leq x\rightarrow z$ linearity: $\left( x\rightarrow y\right) \vee \left( y\rightarrow x\right) =1$ BL: $x\cdot(x\rightarrow y)=x\wedge y$ Remark: The BL identity implies that the lattice is distributive. \end{definition} \begin{definition} A \emph{basic logic algebra} is a \href{FLe-algebras.pdf}{FLe-algebra} $\mathbf{A}=\left\langle A,\vee ,0,\wedge ,1,\cdot ,\rightarrow \right\rangle$ such that linearity: $\left( x\rightarrow y\right) \vee \left( y\rightarrow x\right) =1$ BL: $x\cdot (x\rightarrow y)=x\wedge y$ Remark: The BL identity implies that the identity element $1$ is the top of the lattice. \end{definition} \begin{morphisms} Let $\mathbf{A}$ and $\mathbf{B}$ be basic logic algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x\vee y)=h(x)\vee h(y)$, $h(1)=1$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$ \end{morphisms} \begin{basic_results} \end{basic_results} \begin{examples} \begin{example} \end{example} \end{examples} \begin{table}[h] \begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description}) \begin{tabular}{|ll|}\hline Classtype & variety\\\hline Equational theory & decidable\\\hline Quasiequational theory & \\\hline First-order theory & \\\hline Locally finite & no\\\hline Residual size & unbounded\\\hline Congruence distributive & yes\\\hline Congruence modular & yes\\\hline Congruence n-permutable & yes, $n=2$\\\hline Congruence e-regular & yes, $e=1$\\\hline Congruence uniform & no\\\hline Congruence extension property & yes\\\hline Definable principal congruences & \\\hline Equationally def. pr. cong. & no\\\hline Amalgamation property & \\\hline Strong amalgamation property & \\\hline Epimorphisms are surjective & \\\hline \end{tabular} \end{properties} \end{table} \begin{finite_members} $f(n)=$ number of members of size $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &2\\ f(4)= &5\\ \end{array}$ The number of subdirectly irreducible BL-algebras of size $n$ is $2^{n-2}$. \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \begin{subclasses}\ \href{MV-algebras.pdf}{MV-algebras} \href{Heyting_algebras.pdf}{Heyting algebras} \end{subclasses} \begin{superclasses}\ \href{Generalized_basic_logic_algebras.pdf}{Generalized basic logic algebras} \href{FLew-algebras.pdf}{FLew-algebras} \end{superclasses} \begin{thebibliography}{10} \bibitem{Ln19xx} \end{thebibliography} \end{document} %

http://mathcs.chapman.edu/structuresold/files/Basic_logic_algebras.pdf
%%run pdflatex

%


\documentclass[12pt]{amsart}
\usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref}
\parindent=0pt
\parskip=5pt
\theoremstyle{definition}
\newtheorem{definition}{Definition}
\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
\newtheorem*{examples}{Examples}
\newtheorem{example}{}
\newtheorem*{properties}{Properties}
\newtheorem*{finite_members}{Finite Members}
\newtheorem*{subclasses}{Subclasses}
\newtheorem*{superclasses}{Superclasses}
\newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}}
\markboth{\today}{math.chapman.edu/structures}

\begin{document}
\textbf{\Large Basic logic algebras}

\abbreviation{BLA}

\begin{definition}
A \emph{basic logic algebra} or \emph{BL-algebra} is a structure $\mathbf{A}=\left\langle A,\vee ,0,\wedge ,1,\cdot ,\rightarrow \right\rangle$ such that

$\left\langle A,\vee ,0,\wedge ,1\right\rangle$ is a
\href{Bounded_lattices.pdf}{bounded lattice}

$\left\langle A,\cdot ,1\right\rangle$ is a \href{Commutative_monoids.pdf}{commutative monoid}

$\rightarrow$ gives the residual of $\cdot$:  $x\cdot y\leq z\Longleftrightarrow y\leq x\rightarrow z$

linearity:  $\left( x\rightarrow y\right) \vee \left( y\rightarrow x\right) =1$

BL:  $x\cdot(x\rightarrow y)=x\wedge y$

Remark:
The BL identity implies that the lattice is distributive.
\end{definition}

\begin{definition}
A \emph{basic logic algebra} is a \href{FLe-algebras.pdf}{FLe-algebra} $\mathbf{A}=\left\langle A,\vee ,0,\wedge ,1,\cdot ,\rightarrow \right\rangle$ such that

linearity:  $\left( x\rightarrow y\right) \vee \left( y\rightarrow x\right) =1$

BL:  $x\cdot (x\rightarrow y)=x\wedge y$

Remark:
The BL identity implies that the identity element $1$ is the top of the lattice.
\end{definition}

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be basic logic algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a
homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(1)=1$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0$, $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$
\end{morphisms}

\begin{basic_results}
\end{basic_results}

\begin{examples}
\begin{example}
\end{example}
\end{examples}

\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

\begin{tabular}{|ll|}\hline
Classtype & variety\\\hline
Equational theory & decidable\\\hline
Quasiequational theory & \\\hline
First-order theory & \\\hline
Locally finite & no\\\hline
Residual size & unbounded\\\hline
Congruence distributive & yes\\\hline
Congruence modular & yes\\\hline
Congruence n-permutable & yes, $n=2$\\\hline
Congruence e-regular & yes, $e=1$\\\hline
Congruence uniform & no\\\hline
Congruence extension property & yes\\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong. & no\\\hline
Amalgamation property & \\\hline
Strong amalgamation property & \\\hline
Epimorphisms are surjective & \\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &2\\ f(4)= &5\\ \end{array}$

The number of subdirectly irreducible BL-algebras of size $n$ is $2^{n-2}$.
\end{finite_members}

\hyperbaseurl{http://math.chapman.edu/structures/files/}
\begin{subclasses}\

\href{MV-algebras.pdf}{MV-algebras}

\href{Heyting_algebras.pdf}{Heyting algebras}

\end{subclasses}

\begin{superclasses}\

\href{Generalized_basic_logic_algebras.pdf}{Generalized basic logic algebras}

\href{FLew-algebras.pdf}{FLew-algebras}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
%