Differences

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

bck-meet-semilattices [2010/07/29 15:23] (current)
Line 1: Line 1:
 +=====BCK-meet-semilattices=====
 +Abbreviation: **BCKJMlat**
 +
 +====Definition====
 +A \emph{BCK-meet-semilattice} is a structure $\mathbf{A}=\langle A,\wedge,\rightarrow,1\rangle$ of type $\langle 2,2,0\rangle$ such that
 +
 +(1):  $(x\rightarrow y)\rightarrow
 +((y\rightarrow z)\rightarrow (x\rightarrow z)) = 1$
 +
 +(2):  $1\rightarrow x = x$
 +
 +(3):  $x\rightarrow 1 = 1$
 +
 +(4):  $(x\wedge y)\rightarrow y = 1$
 +
 +(5):  $x\wedge((x\rightarrow y)\rightarrow y) = x$
 +
 +$\wedge$ is idempotent:  $x\wedge x = x$
 +
 +$\wedge$ is commutative:  $x\wedge y = y\wedge x$
 +
 +$\wedge$ is associative:  $(x\wedge y)\wedge z = x\wedge (y\wedge z)$
 +
 +Remark:
 +$x\le y \iff x\rightarrow y=1$ is a partial order, with $1$ as greatest element, and $\wedge$ is a meet in this partial order. [(Idziak1984)]
 +
 +==Morphisms==
 +Let $\mathbf{A}$ and $\mathbf{B}$ be BCK-meet-semilattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
 +
 +$h(x\wedge y)=h(x)\wedge h(y)$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$ and $h(1)=1$.
 +
 +====Examples====
 +Example 1:
 +
 +====Basic results====
 +
 +
 +====Properties====
 +^[[Classtype]]  |variety |
 +^[[Equational theory]]  | |
 +^[[Quasiequational theory]]  | |
 +^[[First-order theory]]  | |
 +^[[Locally finite]]  | |
 +^[[Residual size]]  | |
 +^[[Congruence distributive]]  |yes |
 +^[[Congruence modular]]  |yes |
 +^[[Congruence n-permutable]]  |yes, $n=2$ |
 +^[[Congruence regular]]  | |
 +^[[Congruence uniform]]  | |
 +^[[Congruence extension property]]  | |
 +^[[Definable principal congruences]]  | |
 +^[[Equationally def. pr. cong.]]  | |
 +^[[Amalgamation property]]  | |
 +^[[Strong amalgamation property]]  | |
 +^[[Epimorphisms are surjective]]  | |
 +====Finite members====
 +
 +$\begin{array}{lr}
 +f(1)= &1\\
 +f(2)= &\\
 +f(3)= &\\
 +f(4)= &\\
 +f(5)= &\\
 +f(6)= &\\
 +\end{array}$
 +
 +
 +====Subclasses====
 +[[BCK-lattices]]
 +
 +====Superclasses====
 +[[BCK-algebras]]
 +
 +
 +====References====
 +
 +[(Idziak1984>
 +Pawel M. Idziak, \emph{Lattice operation in BCK-algebras},
 +Math. Japon., \textbf{29}, 1984, 839--846 [[MRreview]]
 +)]