‘Lexical relation’ is a cover term which comprises paradigmatic, in particular conceptual and formal, relations among lexemes.¹ The relevant concepts are introduced in ‘Lexikalische Relationen’.

Semantic relations

Polysemy and homonymy

There are two notorious problems of lexical semantics and lexicography:

• How does one deal with the distinction between homonymy and polysemy?
• How does one represent the related senses ("readings") of a polysemous item?

Both problems find an elegant and satisfactory solution in a lexicon which is implemented as a relational database. Naturally, homonymous items such as

¹pawn goods deposited as security for a loan.
²pawn chessman of the least value.

constitute separate lexical entries just like heteronymous items. Polysemous items are commonly described within the microstructure of one entry. While this seems adequate for print dictionaries, there is a better solution for a lexical database.

At first glance, it might seem sufficient to enumerate the set of senses of a polysemous word in the field dedicated to the meaning definition (field #22 of the microstructure) of one lexical entry. However, a moment's reflection shows that that is not feasible. Polysemy is often related to other lexical differences, for instance in construction, in stylistic status and so on. And each of the senses is provided with its own example. This would lead to an entry structure more or less like this:

It is immediately clear that this would be extremely cumbersome to process both for the human user and for the DBMS.

Next one might think of a nesting infrastructure of the microstructure, where a set of fields that hang together is repeated in the microstructure of an entry. This is permitted in a free-field-structure database and leads to entries like this (the numbers are not in the record, but would be printed):

Polysemous entries in print dictionaries do have such a structure. And some free-field-structure DBMSs like Shoebox/Toolbox are able to handle such structures to a certain extent. However, automatic processing of such records is still difficult; and the solution is not transferable to a relational database. There is, however, a more principled solution. It consists in hypostatizing each such sense of a polysemous item to a record of its own. These lexical entries, of course, have their lemma in common, but differ in the content of their field #22 and other fields.

As the example shows, there is a generic entry, bearing sense n° 0, which has data in all those fields that the various senses have in common. The entries for the different senses, in turn, are not specified for that information, and instead contain data constituting their difference. The relatedness among all these entries is guaranteed by their common lemma (and, if necessary, its homonym number). The specific relation is coded in the sense numbers. In a relational database, these records are linked by these features. When the dictionary is printed, all records that are specified for a sense number are assembled under their lemma with sense number 0, and a structure much like the nesting infrastructure shown above is produced.

This means that polysemous items are treated formally very much like homonymous items. The difference between homonymy and polysemy resides in the fact that the entry of a sense of a polysemous word refers to a mother entry (sense #0) which represents the polysemous item as a whole, while there is no such link among homonyms. This intuitively reflects the nature of the distinction and can easily be revised while elaborating the lexicon.

This solution has the following advantages:

• The distinction between homonymy and polysemy remains flexible. While it does have the status of a categorical distinction in theory, flexibility must be granted in practice.
• The necessity of providing an infrastructure to the various readings of a polysemous items that is much like the corresponding part of the microstructure of an independent (e.g., homonymous) entry is taken care of.
• Some rudimentary kind of object hierarchy with inheritance of properties down the hierarchy is implemented, which may be made use of by an algorithm.

Synonymy

In the monolingual dictionary, synonymy of two entries must be made explicit. This is usually done towards the end of an entry, e.g.:

grasp take hold of sth. with one's hands. Syn.: seize.
...
seize take hold of sth. with one's hands. Syn.: grasp.

What is important here is that the relation is made explicit as such and must not be used in the meaning definition of the items concerned:

grasp seize.
...
seize grasp.

because that would render the definition system circular.

For synonymy in the bilingual dictionary, see the corresponding section.

Hyponomy and hyperonymy

Hyponymy is of particular importance for meanings expressed by nouns such as bird and heron. In principle, one can define hyponymy for adjectival, verbal and even adverbial meanings, too; but that problem may be foregone here.

Hyponymy plays a multiple role in lexicography. First, many lexical entries are in a hyponymy relation. The lexicographer uses this in at least two ways:

• He builds taxonomies headed by such general concepts as ‘animal’, ‘spatial configuration’ etc. These structure lexical fields and, thus, onomasiological dictionaries.
• In defining such meanings that participate in a taxonomy, he follows a definition hierarchy, in the following sense: Since a definition is based on the genus proximum, the immediate hyperonym of a lemma should appear in its definition. Thus (to give a trivial example), mouse is defined as ‘rodent that [differentia specifica_i]’, and hamster is defined as ‘rodent that [differentia specifica_j]’. Only if the lexicographer has elaborated the taxonomy can he make sure to define co-hyponyms in a parallel fashion, e.g. to avoid defining hamster as ‘mammal that [differentia specifica_j]’.

Quite a different problem is posed by taxonomies obtaining in the descriptive vocabulary. For instance, one field of a lexical database specifies the syntactic category of an item; another field specifies the semantic category. Syntactic and semantic categories, however, partake in hyponymy relations. The following, for instance, is a segment of the taxonomy of semantic categories:

1. Perceptual
2. Person
3. Physical
1. artefact
2. celestial body
3. somatic
1. body part
1. animal body part
2. human body part
2. desease
3. function

The hyponymy relation is easily implemented in a relational database, as explained in the corresponding section. With some programming effort, one may then select all the lexical items that belong to a certain category, including any of its subcategories. The DBMS of a free field structure database, however, cannot be programmed in that way. Therefore the hierarchical information must here be expressed by the form of the category names. This is possible as follows:

1. perceptual
2. person
3. physical
1. physical_artefact
2. physical_celestial_body
3. physical_somatic
1. physical_somatic_body_part
1. physical_somatic_body_part_animal
2. physical_somatic_body_part_human
2. physical_somatic_desease
3. physical_somatic_function

(The understrokes are necessary in systems like Toolbox, which would otherwise take the complex terms apart.) Ordering the database according to semantic categories is now a trivial task. One can also select all the items of a category at a given level of the taxonomy by formulating selection conditions like:

• physical_somatic*, which will select all the items of category III.C and any of its subcategories,
• physical_somatic_body_part*, which will select all the items of category III.C.1.

Co-hyponymy – contrast

Cohyponyms may be in a variety of specific relations to each other, of which at least the following are of interest:

• antonymy: e.g. long – short, length – shortness
• complementarity (contradictory contrast): e.g. stallion – mare, alive – dead
• converse relation: e.g. buy – sell, grandparent – grandchild
• minimal contrast: e.g. play – game, who – which

Maintaining these relations in the database helps the lexicographer keep consistency and produce onomasiological reports. As in the case of synonymy, relevant cross-references may be printed at the end of a dictionary article.

Part-whole relation

The part-whole relation is relevant in a few lexical fields such as body parts, plant parts, artefacts. There it may even form a meronomy. The formal treatment of this relation is as for hyponymy.

Derivational relations

A derivational relation is one of word formation. It exists between a complex stem and its components, which are at least one stem – the base – and another component which is a stem in the case of compounding, but some dependent morpheme in the case of derivation proper.

In print dictionaries, derivational relations are often shown by nesting entries that share their base. See the section on macrostructure. In a lexical database, derivational relations between records are independent from the physical order of the records. They consist in mutual links between the complex stem and its two components.

Lexical relations in a database

Logically, a relation is something that connects two or more arguments. In that, it is distinct from a property, which applies to just one argument. One can convert a relation into a property. For instance, let the relation be ‘x is a hyperonym of y’. This may be converted into the notion ‘is a hyperonym of y’, which is a functional property applying to x.

Working with a database allows one to treat relations as relations, not as functional properties. For instance, the lexical entries x = high and y = low are connected by the relation ‘x is the antonym of y’. This is implemented as follows:

Little wonder that the relational database brings out the idea of the relation more clearly than the free-field-structure database. In a relational database, one may then define a “report” that displays, for every lemma, the set of its hyponyms; and similarly for the other relations. This has, unfortunately, no counterpart in the free-field-structure database.

¹ In principle, syntagmatic relations such as selection restrictions could be subsumed under ‘lexical relation’. This need not be discussed here.