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bck-join-semilattices [2010/07/29 15:23] (current)
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+=====BCK-join-semilattices=====
+
+Abbreviation: **BCKJSlat**
+====Definition====
+A \emph{BCK-join-semilattice} is a structure $\mathbf{A}=\langle A,\vee,\rightarrow,1\rangle$ 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):  $x\rightarrow (x\vee y) = 1$
+
+(5):  $x\vee((x\rightarrow y)\rightarrow y) = ((x\rightarrow y)\rightarrow y)$
+
+$\vee$ is idempotent:  $x\vee x = x$
+
+$\vee$ is commutative:  $x\vee y = y\vee x$
+
+$\vee$ is associative:  $(x\vee y)\vee z = x\vee (y\vee z)$
+
+Remark:
+$x\le y \iff x\rightarrow y=1$ is a partial order, with $1$ as greatest element, and $\vee$ is a join
+for this order. [(Idziak1984)]
+
+==Morphisms==
+Let $\mathbf{A}$ and $\mathbf{B}$ be BCK-join-semilattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
+
+$h(x\vee y)=h(x)\vee h(y)$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$ 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]]          | |
+^[[Congruence modular]]               | |
+^[[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====
+
+$\begin{array}{lr} +f(1)= &1\\ +f(2)= &\\ +f(3)= &\\ +f(4)= &\\ +f(5)= &\\ +f(6)= &\\ +\end{array}$
+
+====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]]
+)]\end{document}
+%</pre>