ON THE TERMS “ANABELIOID” AND “FROBENIOID”

Shinichi Mochizuki
September 2015

The purpose of this brief note is to state my position with regard to certain
morphological issues that have been raised by various people concerning the terms“anabelioid” and “Frobenioid” that appear in some of my papers.
(1.) The ﬁrst main point that I would like to make is the following:(P1) Unlike mathematics, the process of determination of “correct ﬁnal answers” with regard to issues of morphology in a language is far from being
an “exact science”.
In particular, although it makes sense to discuss “typical examples” and “generaltrends” in the context of such issues of morphology, in the case of many such issues,
especially those of a relatively subtle nature, it does not make sense — i.e., in
the way that it makes sense in mathematics — to say that a certain form of aword is “right” or “wrong” in any sort of objective sense. Unlike the case withmathematics, the collection of rules that underlie various issues of morphology are
far from being guaranteed to be free of internal logical contradictions. That is tosay, there exist cases where applying various well-accepted rules in diﬀerent ways
gives rise to diﬀerent, or logically inconsistent, ﬁnal answers to a given question— a situation that can never occur in mathematics! This imprecise nature ofissues of morphology leads naturally in discussions of such issues of morphology toconsideration of statistics of occurrence of various patterns of morphology. Thatis to say, languages contain so many “rare exceptions” and “rare inconsistencies”
that the existence of a relatively rare “counterexample” to a given proposition in
discussions of issues of morphology does not carry the same weight as the existenceof a counterexample in a mathematical discussion.
(2.) With regard to the suﬃx “-oid” in English, we ﬁrst review the following basic
point:(P2) The suﬃx “-oid”, which is of Greek origin and means “similar to”, may
be appended to stems of nouns and adjectives.
Examples of words obtained by appending the suﬃx “-oid” to a noun exist in suchgreat abundance that we will not take the time to list examples of this phenomenon
in the present note. Some people have asserted that this suﬃx cannot be appended
to adjectives. First of all, it seems to me that this assertion is amply refuted by theexistence of the following well-known examples:
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SHINICHI MOCHIZUKI· polaroid = (third declension Latin adjective) polar-is + oid,
· australoid = (third declension Latin adjective) austral-is + oid,
· monoid = (ﬁrst/second declension Greek adjective) mon-os + oid.
· aﬃnoid = (third declension Latin adjective) aﬃn-is + oid.
These examples are suﬃciently mainstream that it seems entirely unreasonable todismiss them as “rare cases”. Searches of MathSciNet revealed the following lesswell-known examples used in mathematical papers:· perfectoid = perfect + oid,
· directoid = directed + oid,
· reguloid = regular + oid,
· isoloid = isolated + oid,
· relevantoid = relevant + oid.
Second of all, as a more general matter, the distinction in both Latin and Greek
between nouns and adjectives is often far from clear-cut. That is to say, in both
Latin and Greek, adjectives frequently appear without any explicit noun to modify(typically, in the case of Greek, preceded by a deﬁnite article), in which case theyrefer to “a thing(s)/person(s) which/who is/are ...” and hence function grammatically as nouns. As a result, in many contexts, it becomes quite problematic todetermine where to draw the line between adjectives used in this way and words
that should be regarded as genuine nouns in their own right. This phenomenon
may also be seen in English to a lesser extent in the case of the adjectival suﬃx“-ian” (cf. “Gaussian”, “pfaﬃan”, which typically function as nouns), as well in
the case of the suﬃx “-oid” itself which is frequently used to form both nouns andadjectives.
(3.) Thus, in summary, it follows from (P2) that(P3) it is well within mainstream accepted norms to append the suﬃx “oid” to the stem of the adjective “anabelian” or to the stem of the noun
“Frobenius”.
The nontrivial issue that arises is the question of just what these stems are. Moreover, we note that this issue of determining stems is, in fact, entirely independentof the speciﬁc suﬃx (i.e., such as “-oid”) under consideration. In particular,(P4) statistical issues concerning mainstream morphological patterns withregard to the issue of determination of stems may be investigated by considering suﬃxes other than “-oid” such as, for instance, the more common
suﬃx “-ty”.
The point of view of (P4) will be exploited in the following discussion.
(4.) In the case of the noun “Frobenius”, the issue that arises is the following:(I1) Should the noun “Frobenius” be regarded, from the point of view of issuesof morphology, as(a) a second declension masculine Latin noun, or(b) a non-Greco-Latin word that just happens to end in “s”?
That is to say, the term “Frobenioid” amounts to taking the position of (a), while
the term “Frobeniusoid” (which has been proposed by some people) amounts toON THE TERMS “ANABELIOID” AND “FROBENIOID”

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taking the position of (b). One way to measure the extent to which (a) and (b) aresupported by various mathematical authors is as follows. In the context of Frobenius automorphisms in number theory, (a) corresponds to a plural form “Frobenii”(which also just happens to be the singular genitive form), while (b) corresponds
to a plural form “Frobeniuses”. Then one interesting piece of statistical data is thefollowing. A Google search for “Frobenii”/“Frobeniuses” revealed a sharp gap in
the number of hits:“Frobenii”/“Frobeniuses”: 30700/503(i.e., an overwhelming degree of support for (a)). Such data strongly supports theconclusion that (a) is well within mainstream accepted patterns of usage.
(5.) The issue of determining the stem of the adjective “anabelian” is more subtle.
This subtlety revolves around the following issue:(I2) Is the stem of this adjective “anabelian-”, “anabeli-”, or “anabel-” (i.e.,“-ian-”, “-i-”, or “-”)?
One reason for the subtlety of (I2) is that there does not exist a corresponding noun
“anabelius/-um/-a” (for instance, meaning “anabelian variety”) with stem “-i-”.
On the other hand, although the noun “Abel” (i.e., the name of the mathematician)exists, since the intended meaning of the term “anabelian + oid” is “similar to ananabelian variety” — i.e., not “not (i.e., ‘an-’) similar to the mathematician Abel”!
— the point of view that this noun “Abel” is the appropriate stem is inconsistent
with the semantic aspects of the situation.
(6.) From the point of obtaining statistical data concerning mainstream morphological patterns (cf. (P4)), it is natural to consider (I2) in the context of the following
more general issue:(I3) Is the stem of an adjective of the form “xxx-ian” (i.e., ending in theLatinate suﬃx “-ian”) “xxx-ian-”, “xxx-i-”, or “xxx-” (i.e., “-ian-”, “-i-”,or “-”)?
As discussed in (P4), this issue (I3) may be investigated to a certain extent byconsidering morphology patterns in the case of the suﬃx “-ty” (although, as discussed in (P1), such an approach is by no means completely rigorous in any sortof strict mathematical sense!). For instance, in the case of the adjective “Christian”, the quite common “Christianity” suggests that “-ian-” is one accepted mainstream accepted answer to (I3). On the other hand, the noun “Christity” (thoughcertainly much less common than “Christianity”: a Google search gave rise to114000000/1360 hits for “Christianity”/“Christity”) also exists, hence suggests thatat the very least “-i-” may also be an acceptable answer to (I3). The case of theadjective “Christian” is, however, potentially misleading (not only because of theunusual cultural standing enjoyed by this adjective, but also) since there does in fact
exist a noun with stem “xxx-” (i.e., “Christ”). Similar examples include “barbarian” (where we recall the Greek noun “barbaros” with stem “barbar-”), “Egyptian”(where we recall the Greco-Latin noun “Aegypt-os/-us” (i.e., “Egypt”) with stem“Aegypt-”), “Hungarian” (where we recall the Greek noun “houngroi” (i.e., “Hungarians”) with stem “houngr-”). By contrast, since there does not exist a noun
(i.e., such as a noun “anabel”!) with stem “anabel-”, it seems that, from the pointof view of obtaining statistical data concerning mainstream morphological patterns

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of relevance to (I2), it is more appropriate to consider the following reﬁned versionof (I3):(I4) Consider an adjective of the form “xxx-ian” (i.e., ending in the Latinate
suﬃx “-ian”) for which there does not exist a noun with stem “xxx-” orstem “xxx-i-”. Then is the stem of such an adjective “xxx-ian-”, “xxx-i-”,or “xxx-” (i.e., “-ian-”, “-i-”, or “-”)?
(7.) Although it is not easy to ﬁnd ways to obtain bulk statistical data that addresses(I4) directly, one approach is to consider adjectives obtained by appending theadjectival suﬃx “-arian”. Indeed, in the case of this suﬃx, typically there does notexist a corresponding noun with stem “-ar-” or “-ari-”. Google searches for thesuﬃxes “-ianity”/”-ity”/”-ty” yielded the following (“number of hits”) data:· “humanitar-”: 100/1080/8,· “egalitar-”: 26/471/0,· “utilitar-”: 76/458/1,· “authoritar-”: 28/320/0,· “trinitar-”: 10/306/0,· “totalitar-”: 170/55/2,· “libertar-”: 47/40/43,· “vegetar-”: 418/4/0.
(In this context, it is perhaps of interest to recall examples such as “faculty”,“diﬃculty”, “liberty” of the suﬃx “-ty” (which corresponds to the Latin suﬃx “tas”) appended without an “-i-” to an adjectival stem that ends in a liquid consonant(i.e., “l” or “r”). The English language also contains such words as “cruelty”,“property”, “frailty”, “specialty”, “safety”, “subtlety”, and “entirety”, althoughsuch examples are a bit misleading since their spelling originates from French words
and hence does not necessarily reﬂect the morphological patterns that appear in
the original Latin words.) Again, as discussed in (P1), this data does not suggest
any simple, clear-cut, absolute solutions to (I4). On the other hand, this data does
strongly suggest the following(P5) “bulk statistical answer” to (I4):(a) The answer “-i-” — i.e., which corresponds to “anabelioid” — to(I4) is a very strongly supported acceptable answer to (I4).
(b) The answer “-ian-” — i.e., which corresponds to “anabelianoid” —to (I4) is also a substantially supported acceptable answer to (I4).
(c) The answer “-” — i.e., which corresponds to “anabeloid” — to (I4)
is only relatively rarely supported as an acceptable answer to (I4).
In particular,
this data appears to be amply suﬃcient to refute the assertion that was
been made by some people to the eﬀect that “anabelioid” is incorrect insome sort of objective sense, while “anabeloid” is correct in some sort ofobjective sense.
My own preferences, in light of the bulk statistical answer (P5), may be summarized
as follows:“anabelioid” > “anabelianoid” >>> “anabeloid”.

ON THE TERMS “ANABELIOID” AND “FROBENIOID”

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In particular, I am unable to explicitly endorse the use of the term “anabeloid”
and, moreover, for the reasons explained above, I do not see how this form of
the term can be justiﬁed relative to widely accepted morphology patterns. On
the other hand, again, in accordance with the discussion of (P1), I do not intend
to actively “prosecute” or prohibit the use of the term “anabeloid” (or the term“anabelianoid”).