On he Exac ness and Ca ego ical P ope ies o he
Ca ego y o Basic Hoops
Joaquim Reizi Higuchi
No embe 23, 2025
Abs ac
We conside he ca ego y BH o basic hoops, ha is, in eg al commu a i e
esidua ed monoids sa is ying di isibili y and p elinea i y. As a ini a y a ie y,
BH is known o be comple e, cocomple e, egula and Ba -exac . The aim o
his pape is o gi e a sel -con ained and conc e e desc ip ion o hese ca ego ical
p ope ies in e ms o il e s and cong uences.
We i s e iew he algeb aic s uc u e o basic hoops and desc ibe limi s and
colimi s in BH a he le el o ope a ions. We hen de elop in de ail he co -
espondence be ween il e s and cong uences, and we cha ac e ize il e quo ien s
and ke nel il e s. Using his machine y, we p esen explici il e -based p oo s o
egula i y and exac ness o BH and iden i y in e nal equi alence ela ions wi h
cong uences. We also discuss he ela ion wi h BL-algeb as and o he esidua ed
s uc u es, and we ou line some uni e sal-algeb aic and logical ques ions sugges ed
by his desc ip ion. In his way he pape p o ides a s uc u al no e and pa ial
su ey on he ca ego ical algeb a o basic hoops.
Keywo ds: basic hoops, BL-algeb as, egula ca ego y, Ba -exac ca ego y, il e s, con-
g uences.
MSC 2020: P ima y 06F05; Seconda y 03G10, 18B15, 18E10.
1 In oduc ion
The algeb aic seman ics o subs uc u al and uzzy logics has led o a numbe o impo -
an a ie ies o esidua ed s uc u es, among which hoops and BL-algeb as play a cen al
ole (see, e.g., [5,2,6]). F om he logical poin o iew, hese algeb as p o ide seman ics
o a wide ange o mul iple- alued logics based on con inuous -no ms and hei esidua;
om he algeb aic poin o iew, hey a e in eg al, commu a i e, esidua ed monoids sa -
is ying sui able di isibili y and p elinea i y condi ions. Basic hoops o m a dis inguished
sub a ie y o hoops, ob ained by adding a p elinea i y axiom. They a ise as he implica-
ional educ s o H´ajek’s BL-algeb as, and hus algeb aize he implica ional agmen o
basic logic and o se e al o i s ex ensions, including many s anda d -no m based logics.
While he algeb aic and logical aspec s o hoops and BL-algeb as ha e been ex ensi ely
in es iga ed, he ca ego ical s uc u e o he ca ego y BH o basic hoops and homomo -
phisms has no been analyzed in a sys ema ic way. On he one hand, gene al esul s o
uni e sal algeb a and ca ego ical algeb a imply ha any ini a y a ie y o algeb as o ms
1
a Ba -exac ca ego y in he sense o Ba [1,3]. On he o he hand, in algeb aic logic
and many- alued seman ics i is o en c ucial o wo k wi h explici desc ip ions o limi s,
colimi s, egula epimo phisms and e ec i e equi alence ela ions in e ms o he unde -
lying algeb aic s uc u e, since hese ansla e di ec ly in o he beha iou o deduc i e
il e s, Lindenbaum algeb as, and quo ien logics.
The aim o his pape is o p o ide a de ailed and sel -con ained ca ego ical analysis
o he ca ego y BH, wi h pa icula emphasis on i s ole as an algeb aic seman ics o
basic logic and ela ed many- alued logics. We ocus on he in e ac ion be ween il e s,
cong uences, quo ien s, and he egula and exac s uc u e o BH, and we o mula e
all ou cons uc ions in a il e -based language ha has a di ec logical in e p e a ion
in e ms o deduc i e sys ems. Al hough he inal conclusion ha BH is Ba -exac is
a special case o he gene al heo y o a ie ies, he conc e e il e -based desc ip ion we
ob ain is ailo ed o he many- alued logic se ing and designed o be used as a ool in
he s udy o algeb aizable and -no m-based logics.
Main con ibu ions. The main esul s o he pape can be summa ized as ollows.
(1) We gi e explici cons uc ions o limi s and colimi s in BH. In pa icula , we de-
sc ibe p oduc s and equalize s as ca esian p oduc s and subalgeb as de e mined
by equalizing condi ions, and cop oduc s and coequalize s as ee basic hoops mod-
ulo sui able cong uences. As a consequence, we show ha BH is comple e and
cocomple e, and ha all small limi s and colimi s can be ob ained om p oduc s,
cop oduc s, equalize s and coequalize s (Theo em 3.7). This p o ides a conc e e
desc ip ion o he ca ego ical cons uc ions ha unde lie s anda d model- heo e ic
ope a ions on basic-hoop seman ics.
(2) We es ablish a p ecise il e –cong uence co espondence o basic hoops: il e s on
a basic hoop Aa e in bijec ion wi h cong uences on A, ia explici maps F7→ θF
and θ7→ Fθ(Theo em 4.13). Using his co espondence, we show ha su jec i e
homomo phisms in BH a e exac ly quo ien s by il e s, and we ob ain a il e -based
o mula ion o he i s isomo phism heo em (Theo em 4.19). Logically, his e-
co e s and sys ema izes he co espondence be ween deduc i e il e s, cong uences
and Lindenbaum algeb as o he implica ional agmen o basic logic and i s ex-
ensions.
(3) We cha ac e ize egula epimo phisms in BH as p ecisely he su jec i e homomo -
phisms, and p o e ha hey a e s able unde pullback. Combined wi h he exis ence
o ini e limi s, his shows ha BH is a egula ca ego y in he sense o Ba [1,3]
(Theo em 5.7). F om he iewpoin o algeb aic logic, his yields a clean ac o iza-
ion heo y o seman ic consequence maps be ween basic hoops, wi h ke nel il e s
and image subalgeb as playing he ole o algeb aic in a ian s o deduc i e sys ems.
(4) We gi e an explici desc ip ion o in e nal equi alence ela ions in BH in e ms o
cong uences and il e s, and p o e ha e e y such ela ion is e ec i e: i a ises as
he ke nel pai o he co esponding quo ien homomo phism A→A/θ o A→
A/F. In pa icula , we show ha BH is a Ba -exac ca ego y (Theo em 6.6).
This iden i ies in e nal equi alence ela ions wi h algeb aic coun e pa s o p o able
equi alence in basic logic and ela ed many- alued sys ems.
2
Taken oge he , hese esul s p o ide a conc e e desc ip ion o he egula and exac
s uc u e o BH in algeb aic and il e - heo e ic e ms. F om he poin o iew o alge-
b aic logic and many- alued seman ics, ou analysis ecas s he s anda d co espondence
be ween il e s, cong uences and Lindenbaum algeb as in a ca ego ical language, empha-
sizing he ole o coequalize s, egula epimo phisms, and e ec i e equi alence ela ions
in he algeb aic seman ics o basic logic and i s ex ensions. In his way he pape o e s
a s uc u al amewo k ha can be used in he s udy o algeb aizable mul iple- alued
logics based on basic hoops and hei expansions.
S uc u e o he pape . Sec ion 2collec s basic backg ound on hoops, basic hoops,
and ca ego ical no ions o egula i y and exac ness. In Sec ion 3we cons uc limi s
and colimi s in BH explici ly, and p o e comple eness and cocomple eness. The il e –
cong uence co espondence and he desc ip ion o quo ien s modulo il e s a e de eloped
in Sec ion 4. In Sec ion 5we p o e ha BH is a egula ca ego y, iden i ying egula
epimo phisms wi h su jec i e homomo phisms and s udying hei s abili y unde pull-
back. Sec ion 6con ains he p oo ha BH is Ba -exac : in e nal equi alence ela ions
a e iden i ied wi h cong uences and a e shown o be e ec i e ia hei associa ed il e
quo ien s. Finally, Sec ion 7discusses connec ions wi h BL-algeb as and o he esidua ed
s uc u es, commen s on he logical in e p e a ions o ou esul s, and ou lines se e al
di ec ions o u he in es iga ion in he se ing o mul iple- alued and uzzy logics.
2 P elimina ies
In his sec ion we ix no a ion and ecall he algeb aic and ca ego ical backg ound needed
in he es o he pape . We assume basic amilia i y wi h uni e sal algeb a and ele-
men a y ca ego y heo y. Ou p esen a ion is mainly algeb aic; ca ego ical no ions a e
ecalled only o he ex en needed o desc ibe he s uc u al p ope ies o he ca ego y o
basic hoops ha will be applied la e o algeb aic seman ics o many- alued logics.
Th oughou , we wo k in he algeb aic language
L={·,→,1}
o ype (2,2,0), and we w i e simply · o he monoidal p oduc and → o he esidua ion
ope a ion.
2.1 Basic hoops
We begin by ecalling he no ion o a hoop, and hen specialize o basic hoops.
De ini ion 2.1 (Hoop).Ahoop is an algeb a
A= (A, ·,→,1)
o ype (2,2,0) such ha he ollowing iden i ies hold o all x, y, z ∈A:
(H1) (A, ·,1) is a commu a i e monoid:
x·(y·z)=(x·y)·z, x ·y=y·x, x ·1 = x= 1 ·x;
(H2) x→x= 1;
(H3) (x·y)→z=x→(y→z);
3
(H4) x·(x→y) = y·(y→x).
The equa ional axioma iza ion abo e is s anda d in he li e a u e on hoops; in pa -
icula , i is equi alen o he mo e usual desc ip ion o hoops as cancella i e, in eg al,
commu a i e, esidua ed monoids sa is ying a sui able di isibili y condi ion.
Rema k 2.2 (Na u al o de and esidua ion).Le Abe a hoop. De ine a bina y ela ion
≤on Aby
x≤y:⇐⇒ x→y= 1.
Then:
(i) ≤is a pa ial o de on A;
(ii) (A, ·,1,≤) is an in eg al, commu a i e, esidua ed monoid: o all x, y, z ∈A,
x·y≤z⇐⇒ x≤y→z;
(iii) he iden i y x·(x→y) = y·(y→x) implies ha he ope a ion
x∧y:= x·(x→y) = y·(y→x)
de ines he in imum o xand ywi h espec o ≤, so (A, ∧,≤) is a mee -semila ice.
We call ≤ he na u al o de o A.
In he s anda d many- alued in e p e a ion, he na u al o de compa es u h deg ees,
· ep esen s a o m o conjunc ion based on a -no m, and → ep esen s he co espond-
ing implica ion. The e m-de inable mee will play a ole in ou desc ip ion o il e s,
cong uences and quo ien s, which in u n co espond o deduc i e sys ems and quo ien
logics.
De ini ion 2.3 (Basic hoop).Abasic hoop is a hoop A= (A, ·,→,1) sa is ying he
p elinea i y axiom
(x→y)→z≤((y→x)→z)→z(1)
o all x, y, z ∈A, wi h espec o he na u al o de ≤o Rema k 2.2.
Equi alen ly, since u≤ is he same as u→ = 1, he p elinea i y axiom can be
w i en equa ionally as
(x→y)→z→((y→x)→z)→z= 1,(2)
o all x, y, z ∈A.
We deno e by Ho he class (indeed a ie y) o all hoops, and by BH he class o all
basic hoops.
P oposi ion 2.4. The class BH o basic hoops is a ini a y a ie y o algeb as in he
language L={·,→,1}.
P oo . The iden i ies (H1)–(H4) and (2) o m a ini e se o equa ions in he language L.
Thus BH is an equa ional class in he sense o Bi kho , and is consequen ly closed unde
p oduc s, subalgeb as, and homomo phic images.
4
Rema k 2.5 (Connec ion o BL-algeb as and basic logic).Basic hoops a ise na u ally as
implica ional sub educ s o H´ajek’s BL-algeb as and, mo e gene ally, as he implica ional
educ s o esidua ed s uc u es associa ed wi h con inuous -no ms on [0,1]. The a ie y
BH is gene a ed by such examples and algeb aizes H´ajek’s basic logic a he le el o i s
implica ional agmen . In pa icula , il e s on a basic hoop co espond o deduc i ely
closed se s o implica ions, and quo ien s by il e s yield Lindenbaum basic hoops o
co esponding many- alued logics. In he p esen pape we use his logical iewpoin
mainly as mo i a ion and as an in e p e a ion o ou ca ego ical cons uc ions; no speci ic
p oo sys em will be needed.
2.2 The ca ego y BH o basic hoops
We now pass om he algeb aic s uc u e o basic hoops o hei ca ego ical o ganiza ion,
which will be ou main ool o desc ibing he algeb aic seman ics o basic logic and ela ed
many- alued logics.
De ini ion 2.6 (The ca ego y BH).Le BH deno e he ca ego y de ined as ollows.
•Objec s a e basic hoops A= (A, ·,→,1).
•Mo phisms :A→Ba e homomo phisms o basic hoops, i.e. unc ions :A→B
p ese ing all ope a ions and he cons an :
(x·y) = (x)· (y), (x→y) = (x)→ (y), (1A) = 1B,
o all x, y ∈A.
Composi ion and iden i ies a e aken in he usual way om he ca ego y o se s.
We w i e Se o he ca ego y o se s and unc ions, and deno e by
U:BH −→ Se
he o ge ul unc o sending a basic hoop o i s unde lying se and a homomo phism o
i s unde lying unc ion. The unc o Uis ai h ul and e lec s isomo phisms.
Lemma 2.7 (O de - heo e ic beha iou o mo phisms).Le :A→Bbe a mo phism
in BH. Then:
(i) is mono one wi h espec o he na u al o de s, i.e. x≤Ayimplies (x)≤B (y);
(ii) p ese es he e m-de inable mee :
(x∧y) = (x)∧ (y)
o all x, y ∈A, whe e x∧y:= x·(x→y).
P oo . (i) I x≤Ay, hen x→y= 1A. Applying and using p ese a ion o →and 1
yields
(x)→ (y) = (x→y) = (1A) = 1B,
so (x)≤B (y).
(ii) Using he de ini ion o ∧and p ese a ion o ·and →we ob ain
(x∧y) = x·(x→y)= (x)· (x→y) = (x)· (x)→ (y)= (x)∧ (y),
as equi ed.
5
In he seman ic eading, Lemma 2.7 says ha homomo phisms be ween basic hoops
a e mono one w. . . u h deg ees and p ese e he in e p e a ion o he ( e m-de inable)
mee , so hey beha e as u h-p ese ing maps be ween many- alued models.
Example 2.8 (Te minal and ze o objec ).The one-elemen algeb a 1= ({1},·,→,1), wi h
he only possible de ini ions 1 ·1 = 1 and 1 →1 = 1, is a basic hoop. Fo any basic hoop
A he e is exac ly one homomo phism A→1and exac ly one homomo phism 1→A.
Thus 1is bo h e minal and ini ial in BH, and BH is a poin ed ca ego y.
Since BH is a ini a y a ie y, i inhe i s many s uc u al p ope ies om he gene al
heo y o uni e sal algeb a.
P oposi ion 2.9 (Va ie y- heo e ic p ope ies).The ca ego y BH has he ollowing p op-
e ies:
(i) U:BH →Se c ea es all small limi s and all si ed colimi s;
(ii) BH is comple e and cocomple e: i has all small limi s and colimi s;
(iii) limi s and si ed colimi s in BH a e compu ed on unde lying se s and endowed wi h
he poin wise basic hoop s uc u e.
P oo . This is s anda d o any ini a y a ie y o algeb as. The ca ego y BH is he
Eilenbe g–Moo e ca ego y o he ini a y monad on Se gi en by he ee basic hoop
cons uc ion. Fo such ca ego ies, he o ge ul unc o c ea es all limi s and all si ed
colimi s, and he ca ego y is comple e and cocomple e.
We will no use P oposi ion 2.9 as a black box; ins ead, in Sec ion 3we gi e explici
desc ip ions o p oduc s, equalize s, cop oduc s, and coequalize s in BH, and de i e com-
ple eness and cocomple eness di ec ly om hose cons uc ions. These conc e e desc ip-
ions a e la e in e p e ed in e ms o cons uc ions on algeb aic seman ics o many- alued
logics.
2.3 Regula and exac ca ego ies
We b ie ly ecall he no ions o egula and exac ca ego ies in he sense o Ba . We e e
o s anda d e e ences in ca ego ical algeb a o u he de ails. In ou se ing hey will
be used o o malize he beha iou o quo ien s, ke nel pai s and equi alence ela ions on
basic hoops, and hence o quo ien logics and logical equi alence.
De ini ion 2.10 (Ke nel pai ).Le Cbe a ca ego y wi h pullbacks and le :X→Ybe
a mo phism in C. The ke nel pai o is he pai o mo phisms p1, p2:R→Xob ained
as he pullback o along i sel :
R X
X Y
p2
p1
The objec Ris o en ega ded as an in e nal equi alence ela ion on X, consis ing o
pai s o elemen s iden i ied by .
De ini ion 2.11 (Regula epimo phism).In a ca ego y Cwi h coequalize s, a mo phism
e:X→Yis called a egula epimo phism i i is a coequalize o some pai o mo phisms,
i.e. he e exis 1, 2:W→Xsuch ha
e◦ 1=e◦ 2
and eis uni e sal wi h his p ope y.
6
De ini ion 2.12 (Regula ca ego y).A ca ego y Cis called egula i :
(i) i has all ini e limi s;
(ii) e e y mo phism :X→Yadmi s a ac o iza ion
Xe
−→ Im
−→ Y
whe e eis a egula epimo phism and mis a monomo phism;
(iii) egula epimo phisms a e s able unde pullback: i e:B→Cis a egula epimo -
phism and
P B
A C
p2
p1e
g
is a pullback squa e, hen p1is again a egula epimo phism.
The e a e se e al equi alen o mula ions o egula i y; he one abo e, based on im-
age ac o iza ions and pullback s abili y o egula epimo phisms, is con enien o ou
pu poses.
We nex ecall he ca ego ical no ion o an in e nal equi alence ela ion and o Ba -
exac ness.
De ini ion 2.13 (In e nal equi alence ela ion).Le Cbe a ca ego y wi h ini e limi s and
le Abe an objec o C. An in e nal equi alence ela ion on Aconsis s o a monomo phism
m:R→A×A oge he wi h he induced p ojec ions 1, 2:R→A(ob ained by
composing wi h he i s and second p ojec ions om A×A) such ha :
(i) ( e lexi i y) he diagonal mo phism ∆A:A→A×A ac o s h ough m;
(ii) (symme y)Ris s able unde he lip τ:A×A→A×A,τ(x, y) = (y, x);
(iii) ( ansi i i y) he usual “composi ion” diag am, buil as a pullback o Ro e i sel
and hen mapped back in o R, is well-de ined and sa is ies app op ia e associa i i y
condi ions.
In ui i ely, an in e nal equi alence ela ion is an o dina y equi alence ela ion on he
unde lying objec oge he wi h he equi emen ha i be ealized as a subobjec o
A×A.
De ini ion 2.14 (E ec i e equi alence ela ion).Le Cbe a ca ego y wi h ini e limi s
and coequalize s. An in e nal equi alence ela ion m:R→A×Ais called e ec i e i
he e exis s a mo phism q:A→Qsuch ha Ris (isomo phic o) he ke nel pai o q.
De ini ion 2.15 (Ba -exac ca ego y).A ca ego y Cis called Ba -exac (o simply
exac ) i :
(i) Cis egula ;
(ii) e e y in e nal equi alence ela ion in Cis e ec i e.
I is a classical esul ha any ini a y a ie y o algeb as is Ba -exac . In his
pape we e isi his gene al ac in he pa icula case o basic hoops, gi ing conc e e
cons uc ions o :
•limi s and colimi s in BH (Sec ion 3);
•quo ien s by cong uences and by il e s (Sec ion 4);
•image ac o iza ions and egula epimo phisms (Sec ion 5);
•ke nel pai s and e ec i e equi alence ela ions (Sec ion 6).
7
This no only p o ides an explici algeb aic p oo ha BH is Ba -exac , bu also
cla i ies he ole o il e quo ien s in he ca ego ical s uc u e o basic hoops and, ia he
algeb aic co espondence be ween il e s and heo ies, in he algeb aic seman ics o basic
logic and ela ed many- alued logics.
3 Limi s and colimi s in he ca ego y o basic hoops
In his sec ion we gi e explici desc ip ions o limi s and colimi s in BH. Al hough com-
ple eness and cocomple eness o BH ollow om he gene al heo y o ini a y a ie ies
(see P oposi ion 2.9), we p e e o eco d conc e e cons uc ions o p oduc s, equaliz-
e s, cop oduc s and coequalize s, as hese will be used la e in ou analysis o egula
epimo phisms and exac ness.
3.1 P oduc s, equalize s, and pullbacks
We i s desc ibe p oduc s and equalize s in BH, and hen deduce a conc e e desc ip ion
o pullbacks.
P oposi ion 3.1 (P oduc s).Le {Ai= (Ai,·,→,1i)}i∈Ibe a amily o basic hoops.
De ine he ca esian p oduc o he unde lying se s
A:= Y
i∈I
Ai
and equip Awi h ope a ions and cons an gi en componen wise by
(x·y)(i) := x(i)·y(i),(x→y)(i) := x(i)→y(i),1A(i) := 1i,
o all x, y ∈Aand i∈I. Then:
(a) Ais a basic hoop;
(b) o each i∈I, he p ojec ion
πi:A→Ai, πi(x) := x(i),
is a mo phism in BH;
(c) o any basic hoop Xand any amily o mo phisms i:X→Ai(i∈I), he e exis s
a unique mo phism :X→Asuch ha πi◦ = i o all i∈I.
Consequen ly, Awi h he p ojec ions (πi)i∈Iis he p oduc Qi∈IAiin BH.
P oo . (a) Since each Aisa is ies he de ining equa ions o basic hoops and he ope a ions
on Aa e de ined componen wise, Aalso sa is ies hose equa ions. Thus Ais a basic hoop.
(b) Each πiplainly p ese es ·,→, and 1, hence is a homomo phism.
(c) Gi en i:X→Ai, de ine :X→Aby (x)(i) := i(x). This is a homomo phism
because he homomo phism iden i ies hold coo dina ewise. Uniqueness ollows om he
ac ha any m:X→Awi h πi◦m= imus sa is y m(x)(i) = i(x) o all i, hence
m= .
Equalize s a e ealized as subalgeb as de e mined by poin wise equali ies.
8
P oposi ion 3.2 (Equalize s).Le , g :A→Bbe mo phisms in BH. De ine
E:= {x∈A| (x) = g(x)}.
Then:
(a) Eis a subalgeb a o A, hence a basic hoop wi h he induced ope a ions;
(b) he inclusion e:E ,→Ais he equalize o and gin BH.
P oo . (a) I x, y ∈E, hen
(x·y) = (x)· (y) = g(x)·g(y) = g(x·y),
so x·y∈E. Simila ly,
(x→y) = (x)→ (y) = g(x)→g(y) = g(x→y),
so x→y∈E. Finally, (1A)=1B=g(1A), hence 1A∈E. Thus Eis closed unde he
basic ope a ions and con ains 1, so i is a subalgeb a o A.
(b) By cons uc ion, ◦e=g◦e. Le h:X→Abe any mo phism wi h ◦h=g◦h.
Then o each x∈X, (h(x)) = g(h(x)), hence h(x)∈E. Thus h ac o s uniquely
h ough e ia a homomo phism ¯
h:X→Egi en by ¯
h(x) := h(x) iewed as an elemen
o E. This is he uni e sal p ope y o he equalize .
Pullbacks can be cons uc ed om p oduc s and equalize s in he usual way. In ou
se ing we can desc ibe hem conc e ely as ollows.
Co olla y 3.3 (Pullbacks).Le :X→Zand g:Y→Zbe mo phisms in BH. Le
X×Ybe he p oduc in BH wi h p ojec ions π1:X×Y→Xand π2:X×Y→Y.
Conside he pai o mo phisms
◦π1, g ◦π2:X×Y⇒Z.
Le e:P ,→X×Ybe hei equalize . Then Pcan be iden i ied wi h he subalgeb a
P:= {(x, y)∈X×Y| (x) = g(y)}
o X×Y, equipped wi h he induced ope a ions, and he mo phisms
p1:= π1◦e:P→X, p2:= π2◦e:P→Y
o m a pullback o and gin BH.
P oo . By P oposi ion 3.2, he equalize e:P ,→X×Yo ◦π1and g◦π2is he
subalgeb a o X×Yconsis ing o hose (x, y) such ha (π1(x, y)) = g(π2(x, y)), i.e.
(x) = g(y). Thus Pis exac ly he se displayed in he s a emen , wi h he induced
ope a ions.
Now le h:W→Xand k:W→Ybe mo phisms in BH such ha ◦h=g◦k.
Then he unique mo phism ⟨h, k⟩:W→X×Yin o he p oduc sa is ies
◦π1◦ ⟨h, k⟩= ◦h=g◦k=g◦π2◦ ⟨h, k⟩,
so ⟨h, k⟩equalizes ◦π1and g◦π2and he e o e ac o s uniquely h ough eby a mo phism
u:W→Pwi h e◦u=⟨h, k⟩. By de ini ion o p1and p2, we hen ha e p1◦u=h
and p2◦u=k. Con e sely, since he mo phism u equi ed o p1and p2 o sa is y he
uni e sal p ope y o a pullback is uniquely de e mined by he condi ion e◦u=⟨h, k⟩,
i ollows ha (P, p1, p2) is he pullback o and g.
9
we ha e
(x·(x→y), x ·1) = (x·(x→y), x)∈θ
and
(y·(y→x), y ·1) = (y·(y→x), y)∈θ.
Since x·(x→y) = y·(y→x), hese wo pai s show ha some common elemen
c:= x·(x→y) is θ-equi alen o bo h xand y. By ansi i i y, (x, y)∈θ. Thus θFθ⊆θ,
and he wo inclusions oge he yield θFθ=θ.
Theo em 4.13 (Fil e –cong uence co espondence).Fo a basic hoop A, he assignmen s
F7−→ θF, θ 7−→ Fθ
de ine mu ually in e se, inclusion-p ese ing bijec ions be ween he se o il e s on Aand
he se o cong uences on A.
P oo . By P oposi ions 4.8 and 4.10, he assignmen s a e well-de ined. P oposi ions 4.11
and 4.12 show ha FθF=Fand θFθ=θ, so he wo maps a e mu ually in e se bijec ions.
Mono onici y in each di ec ion is immedia e om he de ini ions.
4.3 Quo ien s modulo il e s
We now use he il e –cong uence co espondence o exp ess cong uence quo ien s in
e ms o il e s, i.e. in e ms o deduc i e sys ems.
De ini ion 4.14 (Quo ien by a il e ).Le Fbe a il e on A. The quo ien o Amodulo
Fis he quo ien basic hoop
A/F := A/θF,
oge he wi h he canonical su jec i e homomo phism
qF:A→A/F, qF(x) := [x]θF.
Lemma 4.15. Fo any il e Fon A,
q−1
F({1A/F }) = F.
P oo . We ha e 1A/F = [1]θFby cons uc ion. Fo any x∈A,
qF(x) = 1A/F ⇐⇒ [x]θF= [1]θF⇐⇒ (x, 1) ∈θF.
As in he p oo o P oposi ion 4.11, his is equi alen o x∈F.
Thus il e s on Aa e exac ly he in e se images o 1 unde homomo phisms in o
quo ien algeb as, ma ching he logical in ui ion ha heo ies a e p eimages o designa ed
u h- alues unde seman ics.
Gi en a mo phism :A→B, we can desc ibe i s ke nel cong uence in e ms o a
il e , namely he p eimage o 1.
De ini ion 4.16 (Ke nel il e ).Le :A→Bbe a mo phism in BH. The ke nel il e
o is
ke 1( ) := {x∈A| (x) = 1B}.
16
Lemma 4.17. Fo any mo phism :A→Bin BH, he subse ke 1( )is a il e on A.
P oo . We ha e 1A∈ke 1( ) because (1A)=1B. I x∈ke 1( ) and x→y∈ke 1( ),
hen (x)=1Band
(x→y) = (x)→ (y) = 1B→ (y) = (y),
so (y) = 1Band y∈ke 1( ). Thus ke 1( ) is a il e .
P oposi ion 4.18 (Ke nel cong uence ia ke nel il e ).Le :A→Bbe a mo phism
in BH. Then
θke 1( )={(x, y)∈A2| (x) = (y)},
i.e. θke 1( )is he ke nel cong uence o .
P oo . I (x) = (y), hen
(x→y) = (x)→ (y) = 1B, (y→x) = (y)→ (x)=1B,
so x→y, y →x∈ke 1( ) and hence xθke 1( )y.
Con e sely, i xθke 1( )y, hen x→y, y →x∈ke 1( ). Thus
(x→y) = 1B, (y→x) = 1B,
which, using p ese a ion o →, gi es
(x)→ (y) = 1B, (y)→ (x)=1B.
Hence (x)≤ (y) and (y)≤ (x) in he na u al o de o B, so (x) = (y). The e o e
θke 1( )={(x, y)| (x) = (y)}.
The usual i s isomo phism heo em o basic hoops can now be ph ased in e ms o
il e quo ien s, making he logical in e p e a ion as quo ien logics by heo ies comple ely
anspa en .
Theo em 4.19 (Fi s isomo phism heo em).Le :A→Bbe a mo phism in BH,
and le F:= ke 1( )be i s ke nel il e . Then:
(a) he map ¯
:A/F →B, ¯
([x]θF) := (x),
is a well-de ined homomo phism;
(b) he image Im( )⊆Bis a subalgeb a o B, and ¯
induces an isomo phism
A/F ∼
=Im( );
(c) in pa icula , i is su jec i e, hen ¯
:A/F →Bis an isomo phism.
P oo . (a) Well-de inedness: i [x]θF= [y]θF, hen (x, y)∈θF=θke 1( )by P oposi-
ion 4.18, so (x) = (y). Thus ¯
is well-de ined. P ese a ion o ·,→, and 1 ollows
om p ese a ion by and he de ini ion o A/F.
(b) The image Im( ) is closed unde he ope a ions o Band con ains 1B, so i is a
subalgeb a. De ine
e:A/F →Im( ), e([x]) := (x).
17
By (a), eis a well-de ined homomo phism and is su jec i e by de ini ion o Im( ). I
e([x]) = e([y]), hen (x) = (y), so (x, y) lies in he ke nel cong uence o , which
equals θFby P oposi ion 4.18. Hence [x] = [y], and eis injec i e. Thus A/F ∼
=Im( ).
(c) I is su jec i e, hen Im( ) = B, and he isomo phism in (b) iden i ies A/F wi h
B.
As an immedia e co olla y, su jec i e mo phisms in BH a e p ecisely quo ien maps
by il e s (up o isomo phism). In logical e ms, e e y su jec i e homomo phism be ween
basic-hoop seman ics a ises as he quo ien by a heo y ( il e ), and con e sely e e y such
quo ien seman ics is induced by a su jec i e homomo phism.
Co olla y 4.20 (Su jec i e homomo phisms as il e quo ien s).Le :A→Bbe a
mo phism in BH. Then is su jec i e i and only i he e exis s a il e Fon Aand an
isomo phism φ:A/F →Bsuch ha
=φ◦qF.
P oo . I is su jec i e, ake F= ke 1( ) and le φ=¯
be he isomo phism o Theo-
em 4.19(c); hen =φ◦qF. Con e sely, i =φ◦qFwi h φan isomo phism, hen qF
is su jec i e and so is .
In pa icula , combined wi h P oposi ion 3.6, his shows ha coequalize s in BH
a e p ecisely quo ien s by il e s (co esponding o he cong uences gene a ed by he
pa allel pai ), so ha he ca ego ical cons uc ion o coequalize s coincides wi h he
logical cons uc ion o quo ien logics by deduc i e heo ies.
5 Regula i y o he ca ego y BH
In his sec ion we p o e ha he ca ego y BH o basic hoops is egula in he sense o
De ini ion 2.12. The key poin s a e:
• egula epimo phisms in BH coincide wi h su jec i e homomo phisms;
•su jec i e homomo phisms a e s able unde pullback;
•e e y mo phism in BH admi s a ac o iza ion as a egula epimo phism ollowed
by a monomo phism.
5.1 Regula epimo phisms in BH
Recall ha in a ca ego y wi h coequalize s, a mo phism is a egula epimo phism i i
is he coequalize o some pai o mo phisms (De ini ion 2.11). In BH, coequalize s a e
ealized as quo ien s by cong uences (P oposi ion 3.6) and, equi alen ly, as quo ien s by
il e s (Co olla y 4.20).
Lemma 5.1. E e y coequalize in BH is a su jec i e homomo phism.
P oo . Le , g :A→Bbe mo phisms in BH and le q:B→Qbe hei coequal-
ize . By P oposi ion 3.6,Qis he quo ien B/θ by he cong uence θgene a ed by
R={( (a), g(a)) |a∈A}, and qis he canonical quo ien map b7→ [b]θ. The un-
de lying unc ion o qis su jec i e by cons uc ion o he quo ien se B/θ, hence qis a
su jec i e homomo phism.
18
Thus e e y egula epimo phism (being a coequalize ) is su jec i e. We now p o e
he con e se.
Lemma 5.2. Le Abe a basic hoop and θa cong uence on A. Le qθ:A→A/θ be he
canonical quo ien . Then he ke nel pai o qθis gi en by he subalgeb a
K:= {(x, y)∈A×A|qθ(x) = qθ(y)}
o A×A, wi h he p ojec ions
k1, k2:K→A, k1(x, y) = x, k2(x, y) = y.
Mo eo e , Kcoincides wi h θas a subse o A×A.
P oo . By de ini ion o he ke nel pai , Kis he pullback o qθalong i sel :
K A
A A/θ
k2
k1qθ
qθ
On unde lying se s, Kis exac ly
{(x, y)∈A×A|qθ(x) = qθ(y)},
and k1, k2a e he es ic ions o he p oduc p ojec ions, hence homomo phisms. The se
Kis clea ly closed unde he ope a ions o A×A, so i is a subalgeb a.
The equali y qθ(x) = qθ(y) is equi alen o [x]θ= [y]θ, which in u n is equi alen o
(x, y)∈θ. Thus Kcoincides wi h θas a subse o A×A.
P oposi ion 5.3 (Su jec i e homomo phisms a e egula epimo phisms).Le :A→B
be a su jec i e mo phism in BH. Then is a egula epimo phism.
P oo . Le
K:= {(x, y)∈A×A| (x) = (y)}
be he ke nel cong uence o , and le q:A→A/K be he canonical quo ien . By
Lemma 5.2, he ke nel pai o qis Kseen as a subalgeb a o A×Awi h p ojec ions
1, 2:K→A, 1(x, y) = x, 2(x, y) = y, and qcoequalizes 1, 2.
De ine u:A/K →Bby u([x]) := (x). This is well de ined: i [x]=[y], hen
(x, y)∈Kand hence (x) = (y). I is a homomo phism since is, and i is su jec i e
because is su jec i e. I u([x]) = u([y]), hen (x) = (y), so (x, y)∈Kand [x]=[y].
Thus uis injec i e and hence an isomo phism A/K ∼
=B. We ha e =u◦q.
We now show ha qis he coequalize o i s ke nel pai ( 1, 2). Ce ainly q◦ 1=q◦ 2
because 1(x, y) and 2(x, y) a e K-equi alen . Le h:A→Cbe any mo phism wi h
h◦ 1=h◦ 2. Then
h(x) = h(y) whene e (x, y)∈K,
so his cons an on K-equi alence classes. De ine ¯
h:A/K →Cby ¯
h([x]) := h(x).
This is well de ined by he abo e and is a homomo phism because his. By cons uc ion,
h=¯
h◦q. I h=¯
h′◦q o ano he ¯
h′:A/K →C, hen ¯
h′([x]) = ¯
h([x]) o all xsince
qis su jec i e. Hence ¯
h′=¯
h. Thus qis he coequalize o ( 1, 2) and hence a egula
epimo phism. Since is he composi e o qwi h an isomo phism, is also a egula
epimo phism.
Combining Lemma 5.1 and P oposi ion 5.3, we ob ain:
Co olla y 5.4 (Regula epimo phisms in BH).In BH, a mo phism is a egula epi-
mo phism i and only i i is a su jec i e homomo phism.
19
5.2 Image ac o iza ions and pullback s abili y
We now show ha e e y mo phism in BH ac o s as a egula epimo phism ollowed by
a monomo phism, and ha egula epimo phisms a e s able unde pullback. Toge he
wi h Theo em 3.7 and De ini ion 2.12, his will gi e egula i y o BH.
P oposi ion 5.5 (Image ac o iza ion).Le :A→Bbe a mo phism in BH. Then
ac o s as
Ae
−→ Im( )m
−→ B,
whe e:
(i) Im( )is he subalgeb a o Bwi h unde lying se { (a)|a∈A};
(ii) e:A→Im( )is a su jec i e homomo phism, hence a egula epimo phism;
(iii) m: Im( ),→Bis he inclusion, which is a monomo phism.
P oo . De ine Im( ) o be he subse { (a)|a∈A}o B. I b1= (a1) and b2= (a2)
a e in Im( ), hen
b1·b2= (a1)· (a2) = (a1·a2)∈Im( ),
and
b1→b2= (a1)→ (a2) = (a1→a2)∈Im( ),
and 1B= (1A)∈Im( ). Thus Im( ) is closed unde he ope a ions and con ains 1B, so
i is a subalgeb a o B.
De ine e:A→Im( ) by e(a) := (a). This is a homomo phism and is su jec-
i e by de ini ion o Im( ). By Co olla y 5.4,eis a egula epimo phism. The inclu-
sion m: Im( ),→Bis a homomo phism ha is injec i e on unde lying se s, hence a
monomo phism in BH. Finally, =m◦eholds by cons uc ion.
We now p o e s abili y o egula epimo phisms unde pullback.
P oposi ion 5.6 (Pullback s abili y o egula epimo phisms).Le
e:B→C
be a egula epimo phism in BH, and le g:A→Cbe any mo phism. Fo m he pullback
squa e
P B
A C
p2
p1e
g
in BH. Then p1:P→Ais a egula epimo phism.
P oo . By Co olla y 5.4,eis a su jec i e homomo phism. The pullback Pcan be ealized
as he subalgeb a
P:= {(a, b)∈A×B|g(a) = e(b)}
o A×B, wi h he induced ope a ions and he p ojec ions p1(a, b) := a,p2(a, b) := b(see
Co olla y 3.3).
We claim ha p1:P→Ais su jec i e. Le a∈Abe a bi a y. Since eis su jec i e,
he e exis s b∈Bwi h e(b) = g(a). Then (a, b)∈Pand p1(a, b) = a. Thus e e y
elemen o Alies in he image o p1, so p1is su jec i e. By Co olla y 5.4,p1is a egula
epimo phism.
20
We can now s a e he main esul o his sec ion.
Theo em 5.7 (Regula i y o BH).The ca ego y BH o basic hoops is a egula ca ego y.
P oo . By Theo em 3.7,BH has all small limi s, and in pa icula all ini e limi s. By
P oposi ion 5.5, e e y mo phism in BH admi s a ac o iza ion as a egula epimo phism
ollowed by a monomo phism. By P oposi ion 5.6, egula epimo phisms a e s able unde
pullback. Thus he condi ions o De ini ion 2.12 a e sa is ied, and BH is egula .
6 Exac ness and e ec i e equi alence ela ions
In his sec ion we p o e ha he egula ca ego y BH is Ba -exac (De ini ion 2.15). By
Theo em 5.7,BH is egula ; i he e o e emains o show ha e e y in e nal equi alence
ela ion in BH is e ec i e, i.e. a ises as he ke nel pai o some mo phism. F om he
poin o iew o algeb aic seman ics o basic logic and ela ed mul iple- alued logics,
his amoun s o showing ha e e y algeb aic no ion o equi alence compa ible wi h he
basic-hoop ope a ions is induced by a quo ien homomo phism, and hence by iden i ying
elemen s ha a e seman ically indis inguishable in an app op ia e quo ien logic.
Because BH is a ini a y a ie y, in e nal equi alence ela ions on a basic hoop Aa e
he same as cong uences on A. We i s make his iden i ica ion explici , and hen show
ha e e y cong uence is he ke nel pai o he co esponding quo ien homomo phism.
In logical e ms, his says ha algeb aic equi alence ela ions a ising in he seman ics o
basic logic a e exac ly hose de e mined by cong uences (o il e s) and he associa ed
Lindenbaum algeb as.
6.1 In e nal equi alence ela ions and cong uences
Le Abe a basic hoop. Recall om De ini ion 2.13 ha an in e nal equi alence ela ion
on Ais a monomo phism
m=⟨ 1, 2⟩:R ,→A×A
such ha he induced s uc u e on Rsa is ies e lexi i y, symme y, and ansi i i y
condi ions in he ca ego ical sense.
Since BH is a a ie y, monomo phisms a e p ecisely injec i e homomo phisms, and
subobjec s o A×Aa e iden i ied wi h subalgeb as o he p oduc basic hoop A×A.
Lemma 6.1. Le Abe a basic hoop.
(i) I θ⊆A×Ais a cong uence on A, hen he inclusion
mθ:θ ,→A×A
( iewing θas a subalgeb a o A×A) oge he wi h he coo dina e p ojec ions 1, 2:
θ→Ade ines an in e nal equi alence ela ion on A.
(ii) Con e sely, i m:R ,→A×Ais an in e nal equi alence ela ion on A, hen i s
image
θ:= m(R)⊆A×A
is a cong uence on A.
21
P oo . (i) I θis a cong uence on A, hen by De ini ion 4.4 i is an equi alence ela ion
on he unde lying se A ha is closed unde he basic ope a ions. Thus θis a subalgeb a
o A×A, and he inclusion mθis a monomo phism in BH. The induced p ojec ions
1, 2:θ→Aa e he es ic ions o he p oduc p ojec ions. The usual se - heo e ic
p ope ies o an equi alence ela ion ( e lexi e, symme ic, ansi i e) ansla e in o he
ca ego ical axioms o an in e nal equi alence ela ion in De ini ion 2.13. Hence mθis an
in e nal equi alence ela ion.
(ii) Con e sely, le m:R ,→A×Abe an in e nal equi alence ela ion. Because
mis a monomo phism in a a ie y, i iden i ies Rwi h a subalgeb a θ⊆A×A ia an
isomo phism R∼
=θo e A×A. The unde lying ela ion θ⊆A×Ais an equi alence
ela ion by he in e nal e lexi i y, symme y, and ansi i i y axioms. Closu e o θunde
he basic ope a ions ollows om he ac ha θis a subalgeb a. Thus θis a cong uence
on A.
P oposi ion 6.2 (In e nal equi alence ela ions s. cong uences).Fo each basic hoop
A, he assignmen
θ7−→ mθ:θ ,→A×A
om cong uences on A o in e nal equi alence ela ions on Ais a bijec ion, wi h in e se
sending an in e nal equi alence ela ion m:R ,→A×A o i s image θ=m(R)⊆A×A.
P oo . Lemma 6.1 shows ha bo h di ec ions a e well-de ined. I we s a om a cong u-
ence θand o m mθ, hen he image o mθis exac ly θi sel . Con e sely, i we s a om
m:R ,→A×Aand le θ=m(R), hen m ac o s as an isomo phism R∼
=θ ollowed
by mθ. Hence he wo assignmen s a e mu ually in e se up o isomo phism o in e nal
equi alence ela ions.
In pa icula , o p o e ha e e y in e nal equi alence ela ion in BH is e ec i e, i
su ices o show ha e e y cong uence θon a basic hoop Ais he ke nel pai o a sui able
mo phism q:A→Q. In logical e ms, hese cong uences co espond o algeb aic
equi alences compa ible wi h he connec i es o he implica ional agmen o basic logic.
6.2 E ec i i y o cong uences
Le Abe a basic hoop and le θbe a cong uence on A. We conside he canonical quo ien
homomo phism
qθ:A→A/θ, qθ(x) := [x]θ,
whe e A/θ is he quo ien basic hoop de ined in Sec ion 4. We show ha θis p ecisely
he ke nel pai o qθ.
Lemma 6.3 (Ke nel pai o he quo ien ).Le θbe a cong uence on Aand qθ:A→A/θ
he canonical quo ien . Then he ke nel pai o qθis he subalgeb a
K:= {(x, y)∈A×A|qθ(x) = qθ(y)}
o A×A, wi h p ojec ions k1, k2:K→Agi en by k1(x, y) = x,k2(x, y) = y. Mo eo e ,
Kcoincides wi h θas a subse o A×A.
22
P oo . By De ini ion 2.10, he ke nel pai o qθis gi en by he pullback o qθalong i sel :
K A
A A/θ
k2
k1qθ
qθ
In BH, his pullback is ealized as he subalgeb a Ko A×Aconsis ing o hose pai s
(x, y) wi h qθ(x) = qθ(y), by Co olla y 3.3. The p ojec ions k1, k2a e ob ained by e-
s ic ing he p oduc p ojec ions and a e homomo phisms.
By de ini ion o he quo ien , qθ(x) = qθ(y) i and only i [x]θ= [y]θ, which is equi alen
o (x, y)∈θ. Thus K=θas subse s o A×A, and since bo h a e subalgeb as wi h he
same unde lying se and ope a ions inhe i ed om A×A, we can iden i y Kand θas
objec s o BH.
P oposi ion 6.4 (Cong uences a e e ec i e).E e y cong uence θon a basic hoop Ais an
e ec i e equi alence ela ion: i is he ke nel pai o he canonical quo ien homomo phism
qθ:A→A/θ.
P oo . By Lemma 6.3, he ke nel pai o qθis gi en by he subobjec θ ,→A×A, iewed
as a subalgeb a o A×Awi h he coo dina e p ojec ions. This is p ecisely he in e nal
equi alence ela ion associa ed wi h θ ia P oposi ion 6.2. Hence θis (isomo phic o) he
ke nel pai o qθand is e ec i e in he sense o De ini ion 2.14.
Combining his wi h he iden i ica ion o in e nal equi alence ela ions and cong u-
ences, we ob ain:
P oposi ion 6.5 (In e nal equi alence ela ions a e e ec i e).E e y in e nal equi alence
ela ion in BH is e ec i e.
P oo . Le Abe a basic hoop and le m:R ,→A×Abe an in e nal equi alence ela ion
on A. By P oposi ion 6.2,mco esponds o a cong uence θon A, and miden i ies R
wi h he subalgeb a θ⊆A×A. By P oposi ion 6.4,θis he ke nel pai o qθ:A→A/θ.
Ke nel pai s a e unique up o isomo phism, so Ris (isomo phic o) he ke nel pai o qθ.
Thus Ris e ec i e.
We can now s a e he main exac ness esul o BH.
Theo em 6.6 (Exac ness o BH).The ca ego y BH o basic hoops is Ba -exac . Tha
is:
(i) BH is a egula ca ego y (Theo em 5.7);
(ii) e e y in e nal equi alence ela ion in BH is e ec i e (P oposi ion 6.5).
P oo . The i s clause is Theo em 5.7, and he second is P oposi ion 6.5. These a e
exac ly he equi emen s o De ini ion 2.15.
Rema k 6.7 (Logical in e p e a ion).The il e –cong uence co espondence (Theo em 4.13)
shows ha e e y in e nal equi alence ela ion on a basic hoop Ais de e mined by a il e
F⊆A, and ha i s quo ien is he il e quo ien A/F. When Ais aken as a Linden-
baum basic hoop o a (possibly implica ional) agmen o basic logic, il e s co espond
o deduc i e sys ems and he quo ien A/F co esponds o a quo ien logic in which
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o mulas a e iden i ied modulo p o able equi alence ela i e o F. In his sense, exac -
ness o BH may be iewed as a ca ego ical e o mula ion o he classical desc ip ion o
cong uence quo ien s in e ms o il e s, and i p o ides a s uc u al explana ion o why
e e y algeb aic no ion o logical equi alence compa ible wi h he ope a ions o a basic
hoop a ises om a quo ien by a deduc i e il e in he many- alued logic se ing.
7 Fu he ema ks and applica ions
The esul s es ablished in he p e ious sec ions show ha he ca ego y BH o basic hoops
is a well-beha ed en i onmen om he poin o iew o ca ego ical algeb a: i is comple e
and cocomple e (Theo em 3.7), egula (Theo em 5.7), and Ba -exac (Theo em 6.6).
Mo eo e , all o hese p ope ies admi conc e e desc ip ions in e ms o il e s, cong u-
ences and quo ien s. In his inal sec ion we collec se e al ema ks and indica e how ou
esul s connec wi h he algeb aic seman ics o mul iple- alued and uzzy logics, as well
as some di ec ions o u he esea ch.
7.1 Connec ions wi h BL-algeb as and ela ed a ie ies
Basic hoops appea na u ally as implica ional sub educ s o BL-algeb as and, mo e gen-
e ally, o a ious classes o commu a i e in eg al esidua ed s uc u es. Le BL deno e
he a ie y o BL-algeb as, in he sense o H´ajek, and conside he o ge ul unc o
Uimp :BL −→ BH
ha sends a BL-algeb a o i s basic hoop educ (A, ·,→,1) and a BL-homomo phism o
i s unde lying homomo phism o basic hoops. In logical e ms, Uimp o ge s all connec i es
excep implica ion and he cons an 1, and hus passes om a ull BL-seman ics o he
implica ional agmen o basic logic and i s ex ensions.
Since BL is a ini a y a ie y, he a gumen s o Sec ions 3–6apply mu a is mu andis
o show ha BL is also comple e, cocomple e, egula , and Ba -exac . In pa icula :
• il e s and cong uences on BL-algeb as co espond bijec i ely;
•su jec i e BL-homomo phisms a e p ecisely he egula epimo phisms;
•in e nal equi alence ela ions on a BL-algeb a a e exac ly i s cong uences, and hey
a e e ec i e.
Thus he ca ego ical pic u e de eloped o basic hoops ex ends o he ull algeb aic seman-
ics o basic logic, and he il e -based desc ip ion o quo ien s and in e nal equi alence
ela ions ca ies o e o BL-algeb as wi hou addi ional echnical complica ions.
F om he iewpoin o many- alued logics based on con inuous -no ms, his means
ha he s anda d cons uc ions on heo ies and quo ien logics (e.g. Lindenbaum algeb as,
conse a i e ex ensions, ac o logics) can be o ganized ca ego ically in e ms o he
egula and exac s uc u e o BL and BH. The unc o Uimp p ese es and e lec s
ini e limi s, and p ese es su jec i e homomo phisms. Hence i is a egula unc o : i
p ese es ini e limi s and egula epimo phisms. As a consequence, much o he egula
and exac s uc u e o BL can be analyzed a he le el o he simple implica ional educ s
and hen anspo ed back along Uimp, p o iding a con enien way o sepa a e hose
phenomena ha a e al eady isible a he implica ional le el om hose ha c ucially
in ol e addi ional connec i es.
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Simila ema ks apply o o he well-known sub a ie ies o hoops and BL-algeb as
(bounded hoops, in olu i e hoops, MV-algeb as, e c.), which p o ide seman ics o im-
po an amilies o mul iple- alued logics. In each case, he a ie y- heo e ic na u e o
he ca ego y gua an ees egula i y and exac ness; he p esen pape sugges s ha i is
use ul o e isi hese ac s wi h explici il e -based desc ip ions o quo ien s and e ec i e
equi alence ela ions in each se ing, wi h an eye owa ds logical applica ions.
7.2 Towa ds homological and p o omodula aspec s
Regula and exac ca ego ies p o ide a con enien amewo k o pa s o homological al-
geb a beyond he abelian con ex , especially in he p esence o addi ional s uc u e such
as p o omodula i y o semi-abelianness. In u n, homological ools in algeb aic ca ego ies
o en yield e ined in a ian s o logical sys ems, o ins ance h ough cen al ex ensions,
commu a o s, and de i ed unc o s associa ed wi h o ge ul unc o s o algeb aic seman-
ics.
I is he e o e na u al o ask o wha ex en he ca ego y BH i s in o he “homologi-
cal” side o ca ego ical algeb a. Recall ha a semi-abelian ca ego y is, oughly speaking,
a poin ed, Ba -exac , p o omodula ca ego y wi h bina y cop oduc s. The ca ego y
BH is poin ed (Example 2.8), Ba -exac (Theo em 6.6), and has all small cop oduc s
(Theo em 3.7). Thus a subs an ial po ion o he semi-abelian axioms is al eady sa -
is ied. Wha emains unclea is whe he BH is p o omodula (in he sense o Bou n)
and whe he sho exac sequences in BH suppo a use ul homological calculus wi h a
meaning ul logical in e p e a ion.
A p esen we do no a emp o se le hese ques ions. Ins ead we eco d he ollowing
p oblems, which a e mo i a ed bo h by ca ego ical conside a ions and by he p ospec o
de eloping “homological” in a ian s o basic logic and ela ed mul iple- alued logics.
(P1) Is he ca ego y BH p o omodula ? I no , can one desc ibe a simple ca ego ical
obs uc ion, possibly ela ed o speci ic logical phenomena (e.g. ailu e o ce ain
in e pola ion o amalgama ion p ope ies) in he associa ed logics?
(P2) Iden i y na u al subclasses o basic hoops ( o example, chains, bounded o in olu-
i e basic hoops) whose ull subca ego ies o BH exhibi s onge homological p op-
e ies, such as p o omodula i y o semi-abelianness, and cla i y he co esponding
logical meaning ( o ins ance, o linea ly o de ed o in olu i e seman ics).
A posi i e answe o (P2) would open he doo o homological echniques (e.g. long
exac sequences, de i ed unc o s) in he s udy o basic hoops and hei logical coun-
e pa s, while a nega i e answe o (P1) would cla i y he p ecise limi a ions o he
ca ego ical s uc u e o BH and o he homological app oach o hese logics.
7.3 Logical in e p e a ions
F om he poin o iew o algeb aic logic, basic hoops p o ide an algeb aic seman ics o
he implica ional agmen o H´ajek’s basic logic and ela ed -no m based logics. In his
con ex , il e s on a basic hoop co espond o deduc i e sys ems, and quo ien s by il e s
a e algeb aic coun e pa s o Lindenbaum algeb as. The ca ego ical esul s o his pape
can he e o e be ead as s uc u al s a emen s abou heo ies and quo ien logics.
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