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—		brouwerian_semilattices [2010/07/29 15:23] (current)
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		+	=====Brouwerian semilattices=====
		+	Abbreviation: **BrSlat**
		+
		+	====Definition====
		+	A \emph{Brouwerian semilattice} is a structure $\mathbf{A}=\langle A, \wedge, 1, \rightarrow\rangle$ such that
		+
		+	$\langle A, \wedge, 1\rangle$ is a [[semilattice with identity]]
		+
		+	$\rightarrow$ gives the residual of $\wedge$:  $x\wedge y\leq z\Longleftrightarrow y\leq x\rightarrow z$
		+
		+	==Morphisms==
		+	Let $\mathbf{A}$ and $\mathbf{B}$ be Brouwerian semilattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a
		+	homomorphism:
		+
		+	$h(x\wedge y)=h(x)\wedge h(y)$, $h(1)=1$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$
		+
		+	====Definition====
		+	A \emph{Brouwerian semilattice} is a [[hoop]] $\mathbf{A}=\langle A, \cdot, 1, \rightarrow\rangle$ such that
		+
		+	$\cdot$ is idempotent:  $x\cdot x=x$
		+
		+	====Examples====
		+	Example 1:
		+
		+	====Basic results====
		+
		+
		+	====Properties====
		+	^[[Classtype]]  |variety |
		+	^[[Equational theory]]  |decidable |
		+	^[[Quasiequational theory]]  | |
		+	^[[First-order theory]]  | |
		+	^[[Locally finite]]  |yes |
		+	^[[Residual size]]  |unbounded |
		+	^[[Congruence distributive]]  |yes |
		+	^[[Congruence modular]]  |yes |
		+	^[[Congruence n-permutable]]  |yes, $n=2$ |
		+	^[[Congruence e-regular]]  |yes, $e=1$ |
		+	^[[Congruence uniform]]  | |
		+	^[[Congruence extension property]]  | |
		+	^[[Definable principal congruences]]  | |
		+	^[[Equationally def. pr. cong.]]  | |
		+	^[[Amalgamation property]]  | |
		+	^[[Strong amalgamation property]]  | |
		+	^[[Epimorphisms are surjective]]  | |
		+
		+	====Finite members====
		+
		+	$\begin{array}{lr}
		+	f(1)= &1\\
		+	f(2)= &1\\
		+	f(3)= &1\\
		+	f(4)= &2\\
		+	f(5)= &3\\
		+	f(6)= &5\\
		+	f(7)= &8\\
		+	f(8)= &15\\
		+	f(9)= &26\\
		+	f(10)= &47\\
		+	f(11)= &82\\
		+	f(12)= &151\\
		+	f(13)= &269\\
		+	f(14)= &494\\
		+	f(15)= &891\\
		+	f(16)= &1639\\
		+	f(17)= &2978\\
		+	f(18)= &5483\\
		+	f(19)= &10006\\
		+	f(20)= &18428\\
		+	\end{array}$
		+
		+	Values known up to size 49 [(ErneHeitzigReinhold2002)]
		+
		+
		+	====Subclasses====
		+	[[Brouwerian algebras]]
		+
		+
		+	====Superclasses====
		+	[[Semilattices with identity]]
		+
		+	[[Hoops]]
		+
		+
		+	====References====
		+
		+	[(ErneHeitzigReinhold2002>
		+	M. Ern\'e, J. Heitzig, J. Reinhold,
		+	\emph{On the number of distributive lattices},
		+	Electronic J. Combinatorics 9 (2002), no. 1, Research Paper 24, 23 pp.
		+	)]