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where <math>s</math> is a schema of length <math>l</math> such that the symbol at position <math>i</math> in <math>s</math> is determined in the following way: if <math>x_i = y_i</math> for all <math>x,y \in A</math> then <math>s_i = x_i</math> otherwise <math>s_i = *</math>. If <math>A = \emptyset</math> then <math>{\downarrow}A = \epsilon_*</math>. One can think of this operator as stacking up all the items in <math>A</math> and if all elements in a column are equivalent, the symbol at that position in <math>s</math> takes this value, otherwise there is a wild card symbol. For example, let <math>A = \{100,000,010\}</math> then <math>{\downarrow}A = **0</math>.
Schemata can be [[partially ordered]]. For any <math>a,b \in \Sigma^l_*</math> we say <math>a \leq b</math> if and only if <math>{\uparrow}a \subseteq {\uparrow}b</math>. It follows that <math>\leq</math> is a [[partial ordering]] on a set of schemata from the [[reflexive operator algebra|reflexivity]], [[Antisymmetric relation|antisymmetry]] and [[transitive relation|transitivity]] of the [[subset]] relation. For example, <math>\epsilon_* \leq 11 \leq 1* \leq **</math>.
This is because <math>{\uparrow}\epsilon_* \subseteq {\uparrow}11 \subseteq {\uparrow}1* \subseteq {\uparrow}** = \emptyset \subseteq \{11\} \subseteq \{11,10\} \subseteq \{11,10,01,00\}</math>.
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== The Schematic Completion and The Schematic Lattice ==
[[File:Schematic Lattice.png|thumb|The Schematic lattice formed from the schematic completion on the set <math>A=\{111, 011, 001\}</math>. Here the schematic lattice <math>(\mathcal{S}(A),\leq)</math> is shown as a [[Hasse diagram]]. ▼
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For a set <math>A \subseteq \Sigma^l</math>, we call the process of calculating the compression on each subset of A, that is <math>\{{\downarrow}X | X \subseteq A\}</math>, the schematic completion of <math>A</math>, denoted <math>\mathcal{S}(A)</math>.<ref name="Fletcher"/>
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The [[poset]] <math>(\mathcal{S}(A),\leq)</math> always forms a [[complete lattice]] called the schematic lattice.
▲[[File:Schematic Lattice.png|thumb|The Schematic lattice formed from the schematic completion on the set <math>A=\{111, 011, 001\}</math>. Here the schematic lattice <math>(\mathcal{S}(A),\leq)</math> is shown as a [[Hasse diagram]].
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The schematic lattice is similar to the concept lattice found in [[Formal concept analysis]].
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{{Reflist}}
{{Evolutionary computation}}
[[Category:Genetic algorithms]]
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