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Revision as of 22:30, 9 February 2009
, 22:30, 9 February 2009→Derived Equivalence Relations: format definition box
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{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black" width="90%"
Definition 12
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
If R c OxSxI,
|- style="height:40px; text-align:center"
| width="80%" |
and x C S,
| width="20%" | <math>\operatorname{Definition~12}</math>
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then the following are identical subsets of O:
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D12a. ROS.x
{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:40px"
D12b. DenR.x
| width="2%" style="border-top:1px solid black" |
| width="18%" style="border-top:1px solid black" | <math>\text{If}\!</math>
D12c. DenR|x
| width="80%" style="border-top:1px solid black" | <math>L ~\subseteq~ O \times S \times I</math>
|- style="height:40px"
D12d. DenR(, x)
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| <math>\text{and}\!</math>
D12e. Den(R, x)
| <math>x ~\in~ S</math>
|- style="height:40px"
D12f. Den(R).x
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| <math>\text{then}\!</math>
D12g. {o C O : <o, x> C Den(R)}
| <math>\text{the following are identical subsets of}~ O \, :</math>
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D12h. {o C O : <o, x, i> C R for some i C I}
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</pre>
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
|- style="height:40px"
| width="2%" style="border-top:1px solid black" |
| width="18%" style="border-top:1px solid black" | <math>\operatorname{D12a.}</math>
| width="80%" style="border-top:1px solid black" | <math>L_{OS} \cdot x</math>
|- style="height:40px"
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| <math>\operatorname{D12b.}</math>
| <math>\operatorname{Den}^L \cdot x</math>
|- style="height:40px"
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| <math>\operatorname{D12c.}</math>
| <math>\operatorname{Den}^L |_x</math>
|- style="height:40px"
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| <math>\operatorname{D12d.}</math>
| <math>\operatorname{Den}^L (-, x)</math>
|- style="height:40px"
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| <math>\operatorname{D12e.}</math>
| <math>\operatorname{Den}(L, x)</math>
|- style="height:40px"
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| <math>\operatorname{D12f.}</math>
| <math>\operatorname{Den}(L) \cdot x</math>
|- style="height:40px"
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| <math>\operatorname{D12g.}</math>
| <math>\{ o \in O ~:~ (o, x) \in \operatorname{Den}(L) \}</math>
|- style="height:40px"
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| <math>\operatorname{D12h.}</math>
| <math>\{ o \in O ~:~ (o, x, i) \in L ~\operatorname{for~some}~ i \in I \}</math>
|}
|}
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