Differences

This shows you the differences between two versions of the page.

implicative_lattices [2010/07/29 15:46] (current)
Line 1: Line 1:
 +=====Implicative lattices=====
 +Abbreviation: **ImpLat**
 +====Definition====
 +An \emph{implicative lattice} is a structure $\mathbf{A}=\langle A,\vee,\wedge,\to\rangle$ such that
 +
 +$\langle A,\vee,\wedge\rangle$ is a [[distributive lattices]]
 +$\to$ is an implication:
 +
 +
 +$x\to(y\vee z) = (x\to y)\vee(x\to z)$
 +
 +
 +$x\to(y\wedge z) = (x\to y)\wedge(x\to z)$
 +
 +
 +$(x\vee y)\to z = (x\to z)\wedge(y\to z)$
 +
 +
 +$(x\wedge y)\to z = (x\to z)\vee(y\to z)$
 +==Morphisms==
 +Let $\mathbf{A}$ and $\mathbf{B}$ be involutive lattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a
 +homomorphism:
 +
 +$h(x\vee y)=h(x)\vee h(y)$, $h(x\vee y)=h(x)\wedge h(y)$, $h(x\to y)=h(x)\to h(y)$
 +
 +Nestor G. Martinez,H. A. Priestley,\emph{On Priestley spaces of lattice-ordered algebraic structures},
 +Order,
 +\textbf{15}1998,297--323[[http://www.ams.org/mathscinet-getitem?mr=2001b:06013|MRreview]]
 +
 +Nestor G. Martinez,\emph{A simplified duality for implicative lattices and $l$-groups},
 +Studia Logica,
 +\textbf{56}1996,185--204[[http://www.ams.org/mathscinet-getitem?mr=97g:06014|MRreview]]
 +
 +====Examples====
 +Example 1:
 +
 +====Basic results====
 +
 +====Properties====
 +^[[Classtype]]  |variety |
 +^[[Equational theory]]  | |
 +^[[Quasiequational theory]]  | |
 +^[[First-order theory]]  | |
 +^[[Locally finite]]  |no |
 +^[[Residual size]]  |unbounded |
 +^[[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]]  | |
 +^[[Strong amalgamation property]]  | |
 +^[[Epimorphisms are surjective]]  | |
 +====Finite members====
 +
 +$\begin{array}{lr}
 +f(1)= &1\\
 +f(2)= &1\\
 +f(3)= &1\\
 +f(4)= &\\
 +f(5)= &\\
 +f(6)= &\\
 +f(7)= &\\
 +f(8)= &\\
 +f(9)= &\\
 +f(10)= &\\
 +\end{array}$
 +
 +====Subclasses====
 +[[Goedel algebras]]
 +
 +[[MV-algebras]]
 +
 +[[Lattice-ordered groups]]
 +
 +====Superclasses====
 +[[Distributive lattices]]
 +
 +
 +====References====
 +
 +[(Ln19xx>
 +)]