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unary_algebras [2010/07/29 15:46]
127.0.0.1 external edit
unary_algebras [2012/07/08 23:34] (current)
jipsen
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-=====Name of class===== +===== Unary Algebras =====
-% Note: replace "Template" with Name_of_class in previous line+
-Abbreviation: **Abbr**+Abbreviation: **Unar**
====Definition==== ====Definition====
-A \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle +A \emph{unary algebra} is a structure $\mathbf{A}=\langle A,(f_i:\in I)\rangle$ of type $\langle 1: i\in I\rangle$ such that 
-...\rangle$ such that +$f_i$ is a unary operation on $A$ for all $i\in I$.
- +
-$\langle A,...\rangle$ is a [[name of class]] +
- +
-$op_1$ is (name of property):  $axiom_1$ +
- +
-$op_2$ is ...:  $...$ +
- +
-Remark: This is a template. +
-If you know something about this class, click on the 'Edit text of this page' link at the bottom and fill out this page. +
- +
-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 unary algebras over the same index set $I$. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:  
-$h(x ... y)=h(x) ... h(y)$+$h(f_i(x))=f_i(h(x))$ for all $i\in I$.
-====Definition==== +====Examples==== 
-An \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle +Example 1: The free unary algebra on one generator is isomorphic to $I^*$, the set of all $n$-tuples of $I$ for $n\in\omega$. The empty tuple is the generator $x$, and the operations $f_i$ are defined by $f_i((i_1,\ldots,i_n))=(i,i_1,\ldots,i_n)$.
-...\rangle$ such that+
-$...$ is ...:  $axiom$ +The free unary algebra on $X$ generators is a union of $|X|$ disjoint copies of the one-generated free algebra.
-   +
-$...$ is ...:  $axiom$ +
- +
-====Examples==== +
-Example 1: +
====Basic results==== ====Basic results====
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====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. 
-^[[Classtype]]                        |(value, see description) [(Ln19xx)]  +^[[Classtype]]                        |variety
-^[[Equational theory]]                | |+^[[Equational theory]]                |undecidable if $|I|>2$ |
^[[Quasiequational theory]]           | | ^[[Quasiequational theory]]           | |
^[[First-order theory]]               | | ^[[First-order theory]]               | |
-^[[Locally finite]]                   | |+^[[Locally finite]]                   |no |
^[[Residual size]]                    | | ^[[Residual size]]                    | |
-^[[Congruence distributive]]          | | +^[[Congruence distributive]]          |no
-^[[Congruence modular]]               | | +^[[Congruence modular]]               |no
-^[[Congruence $n$-permutable]]        | | +^[[Congruence $n$-permutable]]        |no
-^[[Congruence regular]]               | | +^[[Congruence regular]]               |no
-^[[Congruence uniform]]               | | +^[[Congruence uniform]]               |no
-^[[Congruence extension property]]    | | +^[[Congruence extension property]]    |no
-^[[Definable principal congruences]]  | | +^[[Definable principal congruences]]  |no
-^[[Equationally def. pr. cong.]]      | |+^[[Equationally def. pr. cong.]]      |no |
^[[Amalgamation property]]            | | ^[[Amalgamation property]]            | |
^[[Strong amalgamation property]]     | | ^[[Strong amalgamation property]]     | |
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====Finite members==== ====Finite members====
-$\begin{array}{lr} +Depends on $I$
-  f(1)= &1\\ +
-  f(2)= &\\ +
-  f(3)= &\\ +
-  f(4)= &\\ +
-  f(5)= &\\ +
-\end{array}$     +
-$\begin{array}{lr} +
-  f(6)= &\\ +
-  f(7)= &\\ +
-  f(8)= &\\ +
-  f(9)= &\\ +
-  f(10)= &\\ +
-\end{array}$ +
====Subclasses==== ====Subclasses====
-  [[...]] subvariety 
- 
-  [[...]] expansion 
 +[[Permutation unary algebras]] subvariety
====Superclasses==== ====Superclasses====
-  [[...]] supervariety 
-  [[...]] subreduct+[[Duo-unary algebras]] subreduct
====References==== ====References====
- 
-[(Ln19xx> 
-F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 [[MRreview]]  
-)] 
-