No information is lost by dropping the terminal 1s. Thus we may write the following form.
The article linked below tells how forms of that order correspond to a family of digraphs called riffs and a family of graphs called rotes. The riff and rote for 
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A few items of notation are useful in discussing equivalence relations in general and semiotic equivalence relations in particular.
In general, if 
![[x]_E](https://s0.wp.com/latex.php?latex=%5Bx%5D_E&bg=ffffff&fg=000000&s=0&c=20201002)
![[x]](https://s0.wp.com/latex.php?latex=%5Bx%5D&bg=ffffff&fg=000000&s=0&c=20201002)
Thus we have the following definitions.
In the application to sign relations it is useful to extend the square bracket notation in the following ways. If 
![[s]_L](https://s0.wp.com/latex.php?latex=%5Bs%5D_L&bg=ffffff&fg=000000&s=0&c=20201002)
![[s]_L = [s]_{L_{SI}}.](https://s0.wp.com/latex.php?latex=%5Bs%5D_L+%3D+%5Bs%5D_%7BL_%7BSI%7D%7D.&bg=ffffff&fg=000000&s=0&c=20201002)
In many situations there is one further adaptation of the square bracket notation for semiotic equivalence classes which can be useful. Namely, when there is known to exist a particular triple 
![[o]_L](https://s0.wp.com/latex.php?latex=%5Bo%5D_L&bg=ffffff&fg=000000&s=0&c=20201002)
![[s]_L.](https://s0.wp.com/latex.php?latex=%5Bs%5D_L.&bg=ffffff&fg=000000&s=0&c=20201002)
Applying the array of equivalence notations to the sign relations for 
The semiotic equivalence relation for interpreter 
or
In this way the SER for
The semiotic equivalence relation for interpreter 
or
In this way the SER for
Taken all together we have the following picture.
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A semiotic equivalence relation (SER) is a special type of equivalence relation arising in the analysis of sign relations. Generally speaking, any equivalence relation induces a partition of the underlying set of elements, known as the domain or space of the relation, into a family of equivalence classes. In the case of a SER the equivalence classes are called semiotic equivalence classes (SECs) and the partition is called a semiotic partition (SEP).
The sign relations 
A nice property of the sign relations 
Each of the semiotic equivalence relations, 
It’s important to observe the semiotic partitions for interpreters
Information about the contrasting patterns of semiotic equivalence corresponding to the interpreters
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A third aspect of a sign’s complete meaning concerns the relation between its objects and its interpretants, which has no standard name in semiotics. It would be called an induced relation in graph theory or the result of relational composition in relation theory. If an interpretant is recognized as a sign in its own right then its independent reference to an object can be taken as belonging to another moment of denotation, but this neglects the mediational character of the whole transaction in which this occurs. Denotation and connotation have to do with dyadic relations in which the sign plays an active role but here we are dealing with a dyadic relation between objects and interpretants mediated by the sign from an off‑stage position, as it were.
As a relation between objects and interpretants mediated by a sign, this third aspect of meaning may be referred to as the ennotation of a sign and the dyadic relation making up the ennotative aspect of a sign relation 
As it happens, the sign relations 
Tables 5a and 5b show the ennotative components of the sign relations associated with the interpreters 
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Another aspect of a sign’s complete meaning concerns the reference a sign has to its interpretants, which interpretants are collectively known as the connotation of the sign. In the pragmatic theory of sign relations, connotative references fall within the projection of the sign relation on the plane spanned by its sign domain and its interpretant domain.
In the full theory of sign relations the connotative aspect of meaning includes the links a sign has to affects, concepts, ideas, impressions, intentions, and the whole realm of an interpretive agent’s mental states and allied activities, broadly encompassing intellectual associations, emotional impressions, motivational impulses, and real conduct. Taken at the full, in the natural setting of semiotic phenomena, this complex system of references is unlikely ever to find itself mapped in much detail, much less completely formalized, but the tangible warp of its accumulated mass is commonly alluded to as the connotative import of language.
Formally speaking, however, the connotative aspect of meaning presents no additional difficulty. The dyadic relation making up the connotative aspect of a sign relation 
Tables 4a and 4b show the connotative components of the sign relations associated with the interpreters 
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One aspect of a sign’s complete meaning concerns the reference a sign has to its objects, which objects are collectively known as the denotation of the sign. In the pragmatic theory of sign relations, denotative references fall within the projection of the sign relation on the plane spanned by its object domain and its sign domain.
The dyadic relation making up the denotative, referent, or semantic aspect of a sign relation 
Tables 3a and 3b show the denotative components of the sign relations associated with the interpreters 
Looking to the denotative aspects of 
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For an arbitrary triadic relation 
By way of unpacking the set‑theoretic notation, here is what the first definition says in ordinary language.
The dyadic relation resulting from the projection of 
In the case where
Of course, traditions vary with respect to the precise formation and usage of such concepts and terms. Other aspects of meaning have not received their fair share of attention and thus remain innominate in current anatomies of sign relations.
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Soon after I made my third foray into grad school, this time in Systems Engineering, I was trying to explain sign relations to my advisor and he, being the very model of a modern systems engineer, asked me to give a concrete example of a sign relation, as simple as possible without being trivial. After much cudgeling of the grey matter I came up with a pair of examples which had the added benefit of bearing instructive relationships to each other. Despite their simplicity, the examples to follow have subtleties of their own and their careful treatment serves to illustrate important issues in the general theory of signs.
Imagine a discussion between two people, Ann and Bob, and attend only to the aspects of their interpretive practice involving the use of the following nouns and pronouns.
Ann and Bob are not only the passive objects of linguistic references but also the active interpreters of the language they use. The system of interpretation associated with each language user can be represented in the form of an individual three‑place relation known as the sign relation of that interpreter.
In terms of its set‑theoretic extension, a sign relation 
Broadly speaking, the three domains of a sign relation may be any sets at all but the types of sign relations contemplated in formal settings are usually constrained to having 
When it becomes necessary to refer to the whole set of objects and signs in the union of the domains
To facilitate an interest in the formal structures of sign relations and to keep notations as simple as possible as the examples become more complicated, it serves to introduce the following general notations.
Introducing a few abbreviations for use in the Example, we have the following data.
In the present example,
Tables 1a and 1b show the sign relations associated with the interpreters
The Tables codify a rudimentary level of interpretive practice for the agents
Already in this elementary context, there are several meanings which might attach to the project of a formal semiotics, or a formal theory of meaning for signs. In the process of discussing the alternatives, it is useful to introduce a few terms occasionally used in the philosophy of language to point out the needed distinctions. That is the task we’ll turn to next.
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There is a close relationship between the pragmatic theory of signs and the pragmatic theory of inquiry. In fact, the correspondence between the two studies exhibits so many congruences and parallels it is often best to treat them as integral parts of one and the same subject. In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve. In other words, inquiry, “thinking” in its best sense, “is a term denoting the various ways in which things acquire significance” (Dewey, 38).
Tracing the passage of inquiry through the medium of signs calls for an active, intricate form of cooperation between the converging modes of investigation. Its proper character is best understood by realizing the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject the theory of signs is specialized to treat from comparative and structural points of view.
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One of Peirce’s clearest and most complete definitions of a sign is one he gives in the context of providing a definition for logic, and so it is informative to view it in that setting.
Logic will here be defined as formal semiotic. A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time. Namely, a sign is something, A, which brings something, B, its interpretant sign determined or created by it, into the same sort of correspondence with something, C, its object, as that in which itself stands to C.
It is from this definition, together with a definition of “formal”, that I deduce mathematically the principles of logic. I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non‑psychological conception of logic has virtually been quite generally held, though not generally recognized.
— C.S. Peirce, New Elements of Mathematics, vol. 4, 20–21
In the general discussion of diverse theories of signs, the question arises whether signhood is an absolute, essential, indelible, or ontological property of a thing, or whether it is a relational, interpretive, and mutable role a thing may be said to have only within a particular context of relationships.
Peirce’s definition of a sign defines it in relation to its objects and its interpretant signs, and thus defines signhood in relative terms, by means of a predicate with three places. In that definition, signhood is a role in a triadic relation, a role a thing bears or plays in a determinate context of relationships — it is not an absolute or non‑relative property of a thing‑in‑itself, one it possesses independently of all relationships to other things.
Some of the terms Peirce uses in his definition of a sign may need to be elaborated for the contemporary reader.
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Inquiry Into Inquiry 