Mathematical Structures: Basic logic algebras

[Home]Basic logic algebras

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 \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}}
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 \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
\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}
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Last edited June 27, 2004 9:39 am by Peter (diff)
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