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+ | =====Pseudocomplemented distributive lattices===== | ||

+ | Abbreviation: **pcDLat** | ||

+ | |||

+ | ====Definition==== | ||

+ | A \emph{pseudocomplemented distributive lattice} is a structure $\mathbf{L}=\langle L,\vee,0,\wedge,^*\rangle$ such that | ||

+ | |||

+ | |||

+ | $\langle L,\vee,0,\wedge\rangle $ is a [[distributive lattices]] with bottom element $0$ | ||

+ | |||

+ | |||

+ | $x^*$ is the \emph{pseudo complement} of $x$: $y\leq x^* \iff x\wedge y=0$ | ||

+ | |||

+ | ==Morphisms== | ||

+ | Let $\mathbf{L}$ and $\mathbf{M}$ be pseudocomplemented distributive lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a homomorphism: | ||

+ | |||

+ | $h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0$, $h(x^*)=h(x)^*$ | ||

+ | |||

+ | ====Definition==== | ||

+ | A \emph{pseudocomplemented distributive lattice} is a structure $\mathbf{L}=\langle L,\vee,0,\wedge,^*\rangle$ such that | ||

+ | |||

+ | |||

+ | $\langle L,\vee,0,\wedge\rangle $ is a [[distributive lattices]] | ||

+ | |||

+ | |||

+ | $0$ is the bottom element: $0\leq x$ | ||

+ | |||

+ | |||

+ | $x\wedge(x\wedge y)^*=x\wedge y^*$ | ||

+ | |||

+ | |||

+ | $x\wedge 0^*=x$ | ||

+ | |||

+ | |||

+ | $0^{**}=0$ | ||

+ | |||

+ | |||

+ | ====Examples==== | ||

+ | Example 1: | ||

+ | |||

+ | ====Basic results==== | ||

+ | Pseudocomplemented distributive lattices are term equivalent to [[distributive p-algebras]]. | ||

+ | |||

+ | |||

+ | ====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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