Changes

Against this background, other varieties of reflective extension can be specified by means of semantic equations that are considered to be imposed on the elements of <math>F.\!</math>  Taking the reflective extensions <math>\operatorname{Ref}^1 (\text{A})\!</math> and <math>\operatorname{Ref}^1 (\text{B})\!</math> as the first orders of a &ldquo;free&rdquo; project toward reflective closure, variant extensions can be described by relating their entries with those of comparable members in the standard sequences <math>\operatorname{Ref}^n (\text{A})\!</math> and <math>\operatorname{Ref}^n (\text{B}).\!</math>

Against this background, other varieties of reflective extension can be specified by means of semantic equations that are considered to be imposed on the elements of <math>F.\!</math>  Taking the reflective extensions <math>\operatorname{Ref}^1 (\text{A})\!</math> and <math>\operatorname{Ref}^1 (\text{B})\!</math> as the first orders of a &ldquo;free&rdquo; project toward reflective closure, variant extensions can be described by relating their entries with those of comparable members in the standard sequences <math>\operatorname{Ref}^n (\text{A})\!</math> and <math>\operatorname{Ref}^n (\text{B}).\!</math>

A variant pair of reflective extensions, <math>\operatorname{Ref}^1 (\text{A} | E_1)\!</math> and <math>\operatorname{Ref}^1 (\text{B} | E_1),\!</math> are presented in Tables&nbsp;82 and 83, respectively.  These are identical to the corresponding free variants, <math>\operatorname{Ref}^1 (\text{A})\!</math> and <math>\operatorname{Ref}^1 (\text{B}),\!</math> with the exception of those entries that are constrained by the following system of semantic equations.

A variant pair of reflective extensions, <math>\operatorname{Ref}^1 (\text{A} | E_1)\!</math> and <math>\operatorname{Ref}^1 (\text{B} | E_1),\!</math> is presented in Tables&nbsp;82 and 83, respectively.  These are identical to the corresponding free variants, <math>\operatorname{Ref}^1 (\text{A})\!</math> and <math>\operatorname{Ref}^1 (\text{B}),\!</math> with the exception of those entries that are constrained by the following system of semantic equations.

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{| align="center" cellspacing="8" width="90%"

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Another pair of reflective extensions, <math>\operatorname{Ref}^1 (\text{A} | E_2)\!</math> and <math>\operatorname{Ref}^1 (\text{B} | E_2),\!</math> is presented in Tables&nbsp;84 and 85, respectively.  These are identical to the corresponding free variants, <math>\operatorname{Ref}^1 (\text{A})\!</math> and <math>\operatorname{Ref}^1 (\text{B}),\!</math> except for the entries constrained by the following semantic equations.

Another pair of reflective extensions, Ref1(A|E2) and Ref1(B|E2), are presented in Tables&nbsp;84 and 85, respectively.  These are identical to the corresponding "free" variants, Ref1(A) and Ref1(B), except for the entries constrained by the following semantic equations:

E2: <<A>> = A, <<B>> = B, <<i>> = i, <<u>> = u.

</pre>

E_2 :

{}^{\langle\langle} \text{A} {}^{\rangle\rangle} = \text{A},

{}^{\langle\langle} \text{B} {}^{\rangle\rangle} = \text{B},

{}^{\langle\langle} \text{i} {}^{\rangle\rangle} = \text{i},

{}^{\langle\langle} \text{u} {}^{\rangle\rangle} = \text{u}.

\end{matrix}</math>

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