# Differences

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tarski_algebras [2010/07/29 15:46]
127.0.0.1 external edit
tarski_algebras [2015/06/06 11:50] (current)
jipsen
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=====Name of class===== =====Name of class=====
-% Note: replace "Template" with Name_of_class in previous line
-Abbreviation: **Abbr**+Abbreviation: **TarskiA**
====Definition==== ====Definition====
-A \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle +A \emph{Tarski algebra} is a structure$\mathbf{A}=\langle A,\to\rangle$of type$\langle
-...\rangle$such that+2\rangle$ such that $\to$ satisfies the following identities:
-$\langle A,...\rangle$ is a [[name of class]]+$(x\to y)\to x=x$
-$op_1$ is (name of property):  $axiom_1$+$(x\to y)\to y=(y\to x)\to x$
-$op_2$ is ...:  $...$ +$x\to(y\to z)=y\to(x\to z)$
- +
-Remark: This is a template. +
- +
-It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.+
==Morphisms== ==Morphisms==
-Let $\mathbf{A}$ and $\mathbf{B}$ be ... . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:  +Let $\mathbf{A}$ and $\mathbf{B}$ be Tarski algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
-$h(x ... y)=h(x) ... h(y)+$h(x \to y)=h(x) \to h(y)$- + -====Definition==== + -An \emph{...} is a structure$\mathbf{A}=\langle A,...\rangle$of type$\langle +
-...\rangle$such that + - + -$...$is ...:$axiom$+ - + -$...$is ...:$axiom$+ ====Examples==== ====Examples==== -Example 1: +Example 1:$\langle\{0,1\},\to\rangle$where$x\to y=0$iff$x=1$and$y=0\$.
====Basic results==== ====Basic results====
+Tarski algebras are the implication subreducts of Boolean algebras.
====Properties==== ====Properties====
Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described. Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.
-^[[Classtype]]                        |(value, see description) [(Ln19xx)] |+^[[Classtype]]                        | variety |
^[[Equational theory]]                | | ^[[Equational theory]]                | |
^[[Quasiequational theory]]           | | ^[[Quasiequational theory]]           | |