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bck-meet-semilattices [2010/07/29 15:23] (current)
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+=====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]]
+)]