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The connotative component of a sign relation ''L'' can be formalized as its dyadic projection on the plane generated by the sign domain and the interpretant domain, defined as follows:

The connotative component of a sign relation ''L'' can be formalized as its dyadic projection on the plane generated by the sign domain and the interpretant domain, defined as follows:

: ''Con''(''L'') = ''Proj''_''SI'' ''L'' = ''L''_''SI'' = {‹''s'', ''i''› ∈ ''S'' × ''I'' : ‹''o'', ''s'', ''i''› ∈ ''L'' for some ''o'' ∈ ''O''}.

: <math>\operatorname{Con}(L) = \operatorname{proj}_{SI} L = L_{SI} = \{ (s, i) \in S \times I : (o, s, i) \in L ~\text{for some}~ o \in O \}</math>.

The intentional component of semantics for a sign relation ''L'', or its ''second moment of denotation'', is adequately captured by its dyadic projection on the plane generated by the object domain and interpretant domain, defined as follows:

The intentional component of semantics for a sign relation <math>L</math>, or its ''second moment of denotation'', is adequately captured by its dyadic projection on the plane generated by the object domain and interpretant domain, defined as follows:

: ''Int''(''L'') = ''Proj''_''OI''&nbsp;''L'' = ''L''_''OI'' = {‹''o'',&nbsp;''i''› &isin; ''O''&nbsp;&times;&nbsp;''I'' : ‹''o'',&nbsp;''s'',&nbsp;''i''› &isin; ''L'' for some ''s'' &isin; ''S''}.

: <math>\operatorname{Int}(L) = \operatorname{proj}_{OI} L = L_{OI} = \{ (o, i) \in O \times I : (o, s, i) \in L ~\text{for some}~ s \in S \}</math>.

As it happens, the sign relations ''L''_''A'' and ''L''_''B'' in the present example are fully symmetric with respect to exchanging signs and interpretants, so all of the structure of (''L''_''A'')_''OS''&nbsp; and (''L''_''B'')_''OS''&nbsp; is merely echoed in (''L''_''A'')_''OI''&nbsp; and (''L''_''B'')_''OI''&nbsp;, respectively.

As it happens, the sign relations <math>L_\text{A}</math> and <math>L_\text{B}</math> in the present example are fully symmetric with respect to exchanging signs and interpretants, so all of the structure of <math>(L_\text{A})_{OS}</math> and <math>(L_\text{B})_{OS}</math> is merely echoed in <math>(L_\text{A})_{OI}</math> and <math>(L_\text{B})_{OI}</math>, respectively.

'''Note on notation.'''  When there is only one sign relation ''L''_''J''&nbsp; = ''L''(''J'') associated with a given interpreter ''J'', it is convenient to use the following forms of abbreviation:

'''Note on notation.'''  When there is only one sign relation <math>L_J = L(J)</math> associated with a given interpreter <math>J</math>, it is convenient to use the following forms of abbreviation:

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