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+ | =====Distributive p-algebras===== | ||
+ | Abbreviation: **DpAlg** | ||
+ | ====Definition==== | ||
+ | A \emph{distributive p-algebra} is a structure $\mathbf{L}=\langle L,\vee ,0,\wedge ,1,^*\rangle $ such that | ||
+ | |||
+ | |||
+ | $\langle L,\vee,0,\wedge,1\rangle $ is a [[bounded distributive lattices]] | ||
+ | |||
+ | |||
+ | $x^*$ is the \emph{pseudo complement} of $x$: $y\leq x^* \iff x\wedge y=0$ | ||
+ | |||
+ | ==Morphisms== | ||
+ | Let $\mathbf{L}$ and $\mathbf{M}$ be distributive p-algebras. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a | ||
+ | homomorphism: | ||
+ | |||
+ | $h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0 $, $h(1)=1$, $h(x^*)=h(x)^*$ | ||
+ | |||
+ | ====Examples==== | ||
+ | Example 1: | ||
+ | |||
+ | ====Basic results==== | ||
+ | |||
+ | ====Properties==== | ||
+ | ^[[Classtype]] |variety | | ||
+ | ^[[Equational theory]] |decidable | | ||
+ | ^[[Quasiequational theory]] | | | ||
+ | ^[[First-order theory]] | | | ||
+ | ^[[Congruence distributive]] |yes | | ||
+ | ^[[Congruence modular]] |yes | | ||
+ | ^[[Congruence n-permutable]] | | | ||
+ | ^[[Congruence regular]] | | | ||
+ | ^[[Congruence uniform]] | | | ||
+ | ^[[Congruence extension property]] | | | ||
+ | ^[[Definable principal congruences]] | | | ||
+ | ^[[Equationally def. pr. cong.]] | | | ||
+ | ^[[Amalgamation property]] |yes | | ||
+ | ^[[Strong amalgamation property]] | | | ||
+ | ^[[Epimorphisms are surjective]] | | | ||
+ | ^[[Locally finite]] | | | ||
+ | ^[[Residual size]] | | | ||
+ | ====Finite members==== | ||
+ | |||
+ | $\begin{array}{lr} | ||
+ | f(1)= &1\\ | ||
+ | f(2)= &1\\ | ||
+ | f(3)= &1\\ | ||
+ | f(4)= &\\ | ||
+ | f(5)= &\\ | ||
+ | f(6)= &\\ | ||
+ | f(7)= &\\ | ||
+ | \end{array}$ | ||
+ | |||
+ | ====Subclasses==== | ||
+ | [[Distributive double p-algebras]] | ||
+ | |||
+ | ====Superclasses==== | ||
+ | [[Distributive lattices]] | ||
+ | |||
+ | |||
+ | ====References==== | ||
+ | |||
+ | [(Ln19xx> | ||
+ | )] |
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