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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>