Changes

Taken as transition digraphs, <math>\operatorname{Den}^1 (L_\text{A})\!</math> and <math>\operatorname{Den}^1 (L_\text{B})\!</math> summarize the upshots, end results, or effective steps of computation that are involved in the respective evaluations of signs in <math>S\!</math> by <math>\operatorname{Ref}^1 (\text{A})\!</math> and <math>\operatorname{Ref}^1 (\text{B}).\!</math>

Taken as transition digraphs, <math>\operatorname{Den}^1 (L_\text{A})\!</math> and <math>\operatorname{Den}^1 (L_\text{B})\!</math> summarize the upshots, end results, or effective steps of computation that are involved in the respective evaluations of signs in <math>S\!</math> by <math>\operatorname{Ref}^1 (\text{A})\!</math> and <math>\operatorname{Ref}^1 (\text{B}).\!</math>

The connotative components <math>\operatorname{Con}^1 (L_\text{A})\!</math> and <math>\operatorname{Con}^1 (L_\text{B})\!</math> can be viewed as digraphs on the eight points of the syntactic domain <math>S.\!</math> The arcs of these digraphs are given as follows.

The connotative components Con1 (A) and Con1 (B) can be pictured as digraphs on the eight points of the syntactic domain S.  The arcs are given as follows:

1. Con1 (A) inherits from A the structure of a SER on S<1>, having a sling on each of the points in S<1> and two way arcs on the pairs {<A>, <i>} and {<B>, <u>}.  The reflective extension Ref1(A) adds a sling on each point of S<2>, creating a SER on S.

<li><math>\operatorname{Con}^1 (L_\text{A})\!</math> inherits from <math>L_\text{A}\!</math> the structure of a semiotic equivalence relation on <math>S^{(1)},\!</math> having a sling on each point of <math>S^{(1)},\!</math> arcs in both directions between <math>{}^{\langle} \text{A} {}^{\rangle}\!</math> and <math>{}^{\langle} \text{i}{}^{\rangle},\!</math> and arcs in both directions between <math>{}^{\langle} \text{B} {}^{\rangle}\!</math> and <math>{}^{\langle} \text{u}{}^{\rangle}.\!</math> The reflective extension <math>\operatorname{Ref}^1 (L_\text{A})\!</math> adds a sling on each point of <math>S^{(2)},\!</math> creating a semiotic equivalence relation on <math>S.\!</math></li>

2. Con1 (B) inherits from B the structure of a SER on S<1>, having a sling on each of the points in S<1> and two way arcs on the pairs {<A>, <u>} and {<B>, <i>}.  The reflective extension Ref1(B) adds a sling on each point of S<2>, creating a SER on S.

<li><math>\operatorname{Con}^1 (L_\text{B})\!</math> inherits from <math>L_\text{B}\!</math> the structure of a semiotic equivalence relation on <math>S^{(1)},\!</math> having a sling on each point of <math>S^{(1)},\!</math> arcs in both directions between <math>{}^{\langle} \text{A} {}^{\rangle}\!</math> and <math>{}^{\langle} \text{u}{}^{\rangle},\!</math> and arcs in both directions between <math>{}^{\langle} \text{B} {}^{\rangle}\!</math> and <math>{}^{\langle} \text{i}{}^{\rangle}.\!</math> The reflective extension <math>\operatorname{Ref}^1 (L_\text{B})\!</math> adds a sling on each point of <math>S^{(2)},\!</math> creating a semiotic equivalence relation on <math>S.\!</math></li>

Taken as transition digraphs, Con1 (A) and Con1 (B) highlight the associations between signs in Ref1 (A) and Ref1 (B), respectively.

Taken as transition digraphs, <math>\operatorname{Con}^1 (L_\text{A})\!</math> and <math>\operatorname{Con}^1 (L_\text{B})\!</math> highlight the associations between signs in <math>\operatorname{Ref}^1 (L_\text{A})\!</math> and <math>\operatorname{Ref}^1 (L_\text{B}),\!</math> respectively.

The SER given by Con1 (A) for interpreter A has the semantic equations:

The SER given by Con1 (A) for interpreter A has the semantic equations: