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de_morgan_algebras [2010/12/25 13:10]
jipsen
de_morgan_algebras [2012/06/16 02:52] (current)
jipsen
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====Examples==== ====Examples====
Example 1: Let $\{0<a,b<1\}$ be the 4-element lattice with $a,b$ incomparable, and define $'$ by $0'=1,a'=a,b'=b$. Example 1: Let $\{0<a,b<1\}$ be the 4-element lattice with $a,b$ incomparable, and define $'$ by $0'=1,a'=a,b'=b$.
- 
====Basic results==== ====Basic results====
 +
 +The algebra in Example 1 generates the variety of De Morgan algebras, see e.g. http://www.math.uic.edu/~kauffman/DeMorgan.pdf
====Properties==== ====Properties====
-^[[Classtype]]  |Variety +^[[Classtype]]  |variety
-^[[Equational theory]]  | |+^[[Equational theory]]  |decidable |
^[[Quasiequational theory]]  | | ^[[Quasiequational theory]]  | |
^[[First-order theory]]  | | ^[[First-order theory]]  | |
-^[[Congruence distributive]]  |Yes +^[[Congruence distributive]]  |yes
-^[[Congruence modular]]  |Yes |+^[[Congruence modular]]  |yes |
^[[Congruence n-permutable]]  | | ^[[Congruence n-permutable]]  | |
^[[Congruence regular]]  | | ^[[Congruence regular]]  | |
^[[Congruence uniform]]  | | ^[[Congruence uniform]]  | |
-^[[Congruence extension property]]  |Yes |+^[[Congruence extension property]]  |yes |
^[[Definable principal congruences]]  | | ^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]  | | ^[[Equationally def. pr. cong.]]  | |
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^[[Strong amalgamation property]]  | | ^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | | ^[[Epimorphisms are surjective]]  | |
-^[[Locally finite]]  | | +^[[Locally finite]]  |yes
-^[[Residual size]]  | |+^[[Residual size]]  | 4 |
====Finite members==== ====Finite members====
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f(1)= &1\\ f(1)= &1\\
f(2)= &1\\ f(2)= &1\\
-f(3)= &2\\ +f(3)= &1\\ 
-f(4)= &\\ +f(4)= &3\\ 
-f(5)= &\\ +f(5)= &1\\ 
-f(6)= &\\ +f(6)= &4\\ 
-f(7)= &\\ +f(7)= &2\\ 
-f(8)= &\\ +f(8)= &9\\ 
-f(9)= &\\ +f(9)= &5\\ 
-f(10)= &\\+f(10)= &14\\
\end{array}$ \end{array}$
====Subclasses==== ====Subclasses====
-[[Boolean algebras]] +[[Kleene logic algebras]]
====Superclasses==== ====Superclasses====