On skew Heyting algebras - amc-journal.eu

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ARSMATHEMATICACONTEMPORANEA12(2017)37–50
On skew Heyting algebras
KarinCvetko-Vah
UniversityofLjubljana,FacultyofMathematicsandPhysics,
Jadranska19,Ljubljana,Slovenia
Received29October2014,accepted8April2016,publishedonline21April2016
Abstract
InthepresentpaperwegeneralizethenotionofaHeytingalgebratothenon-commuta-
tivesettingandhenceintroducewhatwebelievetobethepropernotionoftheimplication
in skew lattices. We list several examples of skew Heyting algebras, including Heyting
algebras,dualskewBooleanalgebras,conormalskewchainsandalgebrasofpartialmaps
withposetdomains.
Keywords:Skewlattices,Heytingalgebras,non-commutativealgebra,intuitionisticlogic.
Math.Subj.Class.:06F35,03G27
1 Introduction
Non-commutativegeneralizationsoflatticeswereintroducedbyJordan[11]in1949. The
current approach to such objects began with Leech’s 1989 paper on skew lattices [13].
Similarly, skew Boolean algebras are non-commutative generalizations of Boolean alge-
bras. In1936StoneprovedthateachBooleanalgebracanbeembeddedintoafieldofsets
[20]. Likewise, Leech showed in [14, 15] that each right-handed skew Boolean algebra
canbeembeddedintoagenericskewBooleanalgebraofpartialfunctionsfromagivenset
tothecodomain{0,1}. BignallandLeech[5]showedthatskewBooleanalgebrasplaya
centralroleinthestudyofdiscriminatorvarieties.
Though not using categorical language, Stone essentially proved in [20] that the cat-
egory of Boolean algebras and homomorphisms is dual to the category of Boolean topo-
logicalspacesandcontinuousmaps. Generalizationsofthisresultwithinthecommutative
settingyieldPriestleyduality[16,17]betweenboundeddistributivelatticesandPriestley
spaces, and Esakia duality [9] between Heyting algebras and Esakia spaces. (See [4] for
details.)Inarecentpaper[10]onEsakia’swork,GehrkeshowedthatHeytingalgebrasmay
be understood as those distributive lattices for which the embedding into their Booleani-
sation has a right adjoint. A recent line of research generalized the results of Stone and
E-mailaddress:[email protected](KarinCvetko-Vah)
cbThisworkislicensedunderhttp://creativecommons.org/licenses/by/3.0/
38 ArsMath.Contemp.12(2017)37–50
Priestley to the non-commutative setting. By results in [1] and [12], any skew Boolean
algebra is dual to a sheaf of rectangular bands over a locally-compact Boolean space. A
further generalization given in [2] showed that any strongly distributive skew lattice (as
defined below) is dual to a sheaf (of rectangular bands) over a locally compact Priestley
space.
WhileBooleanalgebrasprovidealgebraicmodelsofclassicallogic, Heytingalgebras
provide algebraic models of intuitionistic logic. In the present paper we introduce the
notionofaskewHeytingalgebra. Inpassingtothenon-commutativesettingoneneedsto
sacrificeeitherthetoporthebottomofthealgebrainordernottoendupinthecommutative
setting.Inthepreviouspapers[1],[12]and[2]algebraswithbottomswereconsidered,and
hencethenotionofdistributivitywasgeneralizedtothenotionofso-calledstrongdistribu-
tivity. Ifonetriedtodefineanimplicationoperationinthesettingofstronglydistributive
skewlatticeswithabottomasarightadjointtoconjunction,thatwouldforcetheskewlat-
ticetoalsopossessatopandhencebecommutative,resultinginausualHeytingalgebra.
In order to define implication in the skew lattice setting we consider the ∨−∧ duals of
stronglydistributiveskewlatticeswithabottom,namely,theco-stronglydistributiveskew
latticeswithatop. Thatisnotsurprisingasatopplaysacrucialroleinlogic. Thecategory
ofco-stronglydistributiveskewlatticeswithatopis,ofcourse,isomorphictothecategory
of strongly distributive skew lattices with a bottom. In choosing co-strongly distributive
skew lattices with a top we follow the path paved by Bignall and Spinks in [6], and by
SpinksandVeroffin[19]wheredualskewBooleanalgebraswereintroduced. Forfurther
readingonimplicationsinskewBooleanalgebrasandtheiralgebraicduals,see[7].
After reviewing some preliminary definitions and concepts in Section 2, in the next
sectionweintroducethenotionofaskewHeytingalgebra,provethatsuchalgebrasforma
varietyandshowthatthemaximallatticeimageofaskewHeytingalgebraisageneralized
Heytingalgebra(possiblywithoutabottom).Indeed,aco-stronglydistributiveskewlattice
withatopisthereductofaskewHeytingalgebra,ifandonlyifitsmaximallatticeimage
formsageneralizedHeytingalgebra. (SeeTheorem3.5.) Thisleadstoanumberofuseful
corollaries and examples. We finish with Section 4 where we explore the consequences
of our results to duality theory, and describe how skew Heyting algebras correspond to
sheavesoverlocalEsakiaspaces.
2 Preliminaries
A skew lattice is an algebra S = (S;∧,∨) of type (2,2) such that ∧ and ∨ are both
idempotentandassociativeandtheysatisfythefollowingabsorptionlaws:
x∧(x∨y)=x=x∨(x∧y)and(x∧y)∨y =y =(x∨y)∧y.
Theseidentitiesarecollectivelyequivalenttothepairofequivalences:x∧y =x⇔x∨y =
yandx∧y =y ⇔x∨y =x.
OnaskewlatticeSonecandefinethenaturalpartialorderbystatingthatx≤yifand
onlyifx∨y = y = y∨x,orequivalentelyx∧y = z = y∧x,andthenaturalpreorder
byx(cid:22)yifandonlyify∨x∨y =y,orequivalentelyx∧y∧x=x. Green’sequivalence
relationDisthendefinedby
xDyifandonlyifx(cid:22)yandy (cid:22)x. (2.1)
K.Cvetko-Vah:OnskewHeytingalgebras 39
Lemma 2.1. ([8]). For elements x and y elements of a skew lattice S the following are
equivalent:
(i) x≤y,
(ii) x∨y∨x=y,
(iii) y∧x∧y =x.
Leech’s First Decomposition Theorem for skew lattices states that the relation D is a
congruenceonaskewlatticeS,S/D isthemaximallatticeimageofS,andeachcongru-
ence class is a maximal rectangular skew lattice in S [13]. Rectangular skew lattices are
characterizedbyx∧y∧z = x∧z, orequivalentelyx∨y∨z = x∨z. Wedenotethe
D-classcontaininganelementxbyD .
x
Itwasprovedin[13]thataskewlatticealwaysformsaregularband foreitherofthe
operations∧,∨,i.e. itsatisfiestheidentities
x∧u∧x∧v∧x=x∧u∧v∧xandx∨u∨x∨v∨x=x∨u∨v∨x.
Askewlatticewithtopisanalgebra(S;∧,∨,1)oftype(2,2,0)suchthat(S;∧,∨)is
askewlatticeand x∨1 = 1 = 1∨x, orequivalently x∧1 = x = 1∧x, holdsforall
x ∈ S. Askewlatticewithbottomisdefinedduallyandthebottom,ifitexists,isusually
denotedby0.
Furthermore, a skew lattice is called strongly distributive if it satisfies the following
identities:
x∧(y∨z)=(x∧y)∨(x∧z)and(x∨y)∧z =(x∧z)∨(y∧z);
anditiscalledco-stronglydistributiveifitsatisfiestheidentities:
x∨(y∧z)=(x∨y)∧(x∨z)and(x∧y)∨z =(x∨z)∧(y∨z).
If a skew lattice S is either strongly distributive or co-strongly distributive then S is
distributiveinthatitsatisfiestheidentities
x∧(y∨z)∧x=(x∧y∧x)∨(x∧z∧x)andx∨(y∧z)∨x=(x∨y∨x)∧(x∨z∨x).
A skew lattice S that is jointly strongly distributive and co-strongly distributive is bi-
normal, i.e. S factors as a direct product of a lattice L and a rectangular skew lattice B,
S∼=L×B,withLinthiscasebeingdistributive. (See[15]and[18].)
ApplyingdualitytoaresultofLeech[15],itfollowsthataskewlatticeSisco-strongly
distributiveifandonlyifSisjointly:
• quasi-distributive: themaximallatticeimageS/Disadistributivelattice,
• symmetric: x∧y =y∧xifandonlyifx∨y =y∨x,and
• conormal: x∨y∨z∨w =x∨z∨y∨w.
Ifaskewlatticeisconormalthengivenanyu∈S theset
u↑={u∨x∨u|x∈S}={x∈S|u≤x}
formsa(commutative)latticefortheinducedoperations∧and∨,cf. [15].
Thefollowinglemmaisthedualofawellknownresultinskewlatticetheory.
40 ArsMath.Contemp.12(2017)37–50
Lemma 2.2. Let S be a conormal skew lattice and let A and B be D-classes such that
B ≤AholdsinthelatticeS/D. Givenb∈Bthereexistsauniquea∈Asuchthatb≤a.
Proof. Firsttheuniqueness. Ifaanda(cid:48) bothsatisfythedesiredpropertythenbyLemma
2.1 we have a = b∨a∨b and likewise a(cid:48) = b∨a(cid:48) ∨b. Now, using idempotency of ∨,
conormalityandthefactthataDa(cid:48)weobtain:
a=b∨a∨b=b∨a∨a(cid:48)∨a∨b=
b∨a∨a(cid:48)∨b=b∨a(cid:48)∨a∨a(cid:48)∨b=b∨a(cid:48)∨b=a(cid:48).
Toprovetheexistenceofatakeanyx ∈ Aandseta = b∨x∨b. Thena ∈ Aandusing
theidempotencyof∨weget:
b∨a∨b=b∨(b∨x∨b)∨b=b∨x∨b=a
whichimpliesb≤a.
Animportantclassofstronglydistributiveskewlatticesthathaveabottomistheclass
of skew Boolean algebras where by a skew Boolean algebra we mean an algebra S =
(S;∧,∨,\,0)where(S;∧,∨,0)isastronglydistributiveskewlatticewithbottom0,and
\isabinaryoperationonSsuchthatboth(x∧y∧x)∨(x\y)=x=(x\y)∨(x∧y∧x)
and(x∧y∧x)∧(x\y) = 0 = (x\y)∧(x∧y∧x). Givenanyelementuofaskew
BooleanalgebraStheset
u↓={u∧x∧u|x∈S}={x∈S|x≤u}
isaBooleanalgebrawithtopuandwithu\xbeingthecomplementofu∧x∧uinu↓.
RecallthataHeytingalgebraisanalgebraH=(H;∧,∨,→,1,0)suchthat(H,∧,∨,
1,0)isaboundeddistributivelatticethatsatisfiesthecondition:
(HA) x∧y ≤ziffx≤y →z.
Stated otherwise, ∀y,z ∈ H the sublattice {x ∈ H|x∧y ≤ z} is nonempty and con-
tains a top element to be denoted by y → z. Thus, given a bounded distributive lattice
(H;∧,∨,1,0),ifabinaryoperation→existsthatmakes(H;∧,∨,→,1,0)aHeytingal-
gebra,thenitisuniquebecauseitisalreadythereimplicitly. Indeed,giventwoisomorphic
lattices,ifeitheristhelatticereductofaHeytingalgebrathensoistheother,andbothare
isomorphicasHeytingalgebras.
Equivalently,(HA)canbereplacedbythefollowingsetofidentities:
(H1) (x→x)=1,
(H2) x∧(x→y)=x∧y,
(H3) y∧(x→y)=y,
(H4) x→(y∧z)=(x→y)∧(x→z).
Lemma2.3. InanyHeytingalgebra,x→y =(x∨y)→y.
AgeneralizedHeytingalgebraisanalgebraA = (A;∧,∨,→,1)suchthatthereduct
(A,∧,∨,1) is a distributive lattice with top 1, and condition (HA) holds. If it also has a
bottom, it is a Heyting algebra. In general, each upset u↑ forms a Heyting algebra. The
identitiesabovealsocharacterizethismoregeneralclassofalgebras,whichareoftencalled
Brouwerianalgebras.
K.Cvetko-Vah:OnskewHeytingalgebras 41
3 SkewHeytingalgebras
AskewHeytinglatticeisanalgebraS=(S;∧,∨,1)oftype(2,2,0)suchthat:
• (S;∧,∨,1)isaco-stronglydistributiveskewlatticewithtop1. Eachupsetusupis
thusaboundeddistributivelattice.
• foranyu ∈ S anoperation→ canbedefinedonu↑suchthat(u↑;∧,∨,→ ,1,u)
u u
isaHeytingalgebrawithtop1andbottomu.
GivenaskewHeytinglatticeS,define→onS bysetting
x→y =(y∨x∨y)→ y.
y
A skew Heyting algebra is an algebra S = (S;∧,∨,→,1) of type (2,2,2,0) such that
(S;∧,∨,1) is a skew Heyting lattice and → is the implication thus induced. A sense of
globalcoherencefor→onS isgivenby:
Lemma3.1. LetSbeaskewHeytinglatticewith→asdefinedaboveandletx,y,u ∈ S
besuchthatbothx,y ∈u↑hold. Thenx→y =x→ y.
u
Proof. As x and y both lie in u↑, they commute. By the definition of →, x → y =
(x∨y) → y ≥ y by (H3). On the other hand, since → is the Heyting implication in
y u
the Heyting algebra u↑ it follows that x → y = (x∨y) → y ≥ y. Thus y, x∨y,
u u
(x∨y) → y and(x∨y) → y alllieiintheHeytingalgebray↑. Themaximalelement
y u
characterizationofboth(x∨y)→ yand(x∨y)→ yforcesbothtoagree.
y u
WewillusetheaxiomsofHeytingalgebrastoderiveanaxiomatizationofskewHeyting
algebras. ThereadershouldfindmostoftheaxiomsofTheorem3.2belowtobeintuitively
cleargeneralizationstothenon-commutativecase. However,twoaxiomsshouldbegiven
furtherexplanation. Firstly,theuinaxiom(SH4)belowappearssinceuponpassingtothe
non-commutative case, an element that is both below x and y with respect to the partial
order≤nolongerneedexist.(Wecanhavex∧y∧x≤xbutnotx∧y∧x≤yingeneral.)
Similarly, axiom(SH0)isneededsinceinthenon-commutativecaseitnolongerfollows
fromtheotheraxioms,thereasonbeingthatingeneralx≤y∨x∨yneednothold.
Theorem 3.2. Let (S;∧,∨,→,1) be analgebra of type (2,2,2,0) such that (S;∧,∨,1)
isaco-stronglydistributiveskewlatticewithtop1. Then(S;∧,∨,→,1)isaskewHeyting
algebraifandonlyifitsatisfiesthefollowingaxioms:
(SH0) x→y =(y∨x∨y)→y.
(SH1) x→x=1,
(SH2) x∧(x→y)∧x=x∧y∧x,
(SH3) y∧(x→y)=yand(x→y)∧y =y,
(SH4) x→(u∨(y∧z)∨u)=(x→(u∨y∨u))∧(x→(u∨z∨u)).
Proof. AssumethatSisaskewHeytingalgebra.
(SH0). Bothx → yand(y∨x∨y) → yaredefinedas(y∨x∨y) → y. Thusthey
y
areequal.
(SH1). Thisistruebecause1∧x=xistrueinx↑.
42 ArsMath.Contemp.12(2017)37–50
(SH2). Iny↑(H2)implies(y∨x∨y)∧((y∨x∨y) → y) = (y∨x∨y)∧y = y.
y
Hence
x∧(y∨x∨y)∧(x→y)∧x=x∧y∧x.
Ontheotherhand,
x∧(y∨x∨y)∧(x→y)∧x=x∧(y∨x∨y)∧x∧(x→y)∧x=x∧(x→y)∧x,
wherewehaveusedtheregularityof∧andthefactthatx(cid:22)y∨x∨y.
(SH3). Both identities hold because y ∧ (y ∨ x ∨ y) = y in y↑. Thus x → y =
(y∨x∨y)→y ≥y.
(SH4). Firstnotethat(SH4)isequivalentto
(SH4’)(u∨x∨u)→(u∨(y∧z)∨u)=((u∨x∨u)→(u∨y∨u))∧((u∨x∨u)→
(u∨z∨u)).
Indeed,(SH0)andtheconormalityof∨giveboth
(u∨x∨u)→(u∨w∨u)=(u∨x∨w∨u)→(u∨w∨u)
and
x→(u∨w∨u)=(u∨x∨w∨u)→(u∨w∨u)
sothat
x→(u∨w∨u)=(u∨x∨u)→(u∨w∨u).
Henceitsufficestoprovethat(SH4’)holds.
Observethatdistributivityimplies
(u∨y∨u)∧(u∨z∨u)=u∨(y∧z)∨u. (3.1)
Since u∨x∨u, u∨y ∨u, u∨z ∨u and u∨(y ∧z)∨u all lie in u↑ we can simply
computeinu↑. Using(3.1)andaxiom(H4)forHeytingalgebrasweobtain:(u∨x∨u)→
(u∨(y ∧z)∨u) = (u∨x∨u) → ((u∨y ∨u)∧(u∨z ∨u)) = ((u∨x∨u) →
(u∨y∨u))∧((u∨x∨u)→(u∨z∨u)).
To prove the converse assume that (SH0)–(SH4) hold. Given arbitrary u ∈ S and
x,y,z ∈u↑setx→ y =x→y. Axiom(SH3)impliesthatx→y ∈y↑⊆u↑. Thusthe
u
restriction→ of→tou↑iswelldefined. Sinceu↑iscommutativewithbottomu,axioms
u
(SH1)–(SH4)for→respectivelysimplifyto(H1)–(H4)for→ , making→ theHeyting
u u
implication on u↑. Axiom (SH0) assures that → is indeed the skew Heyting implication
satisfyinga→b=(b∨a∨b)→ b,foranya,b∈S.
b
Corollary3.3. SkewHeytingalgebrasformavariety.
In the remainder of the paper, given a skew Heyting algebra we shall simplify the
notation→ to→whenreferringtotheHeytingimplicationinu↑. Remarksmadeabout
u
HeytingalgebrasinSection2applyherealso. Givenaco-stronglydistributiveskewlattice
(S;∧,∨,1)withatop1,ifabinaryoperation→existsthatmakes(S;∧,∨,→,1)askew
Heyting algebra, then it is unique since it is already there implicitly. Hence, given two
isomorphic skew lattices, if either is the reduct of a skew Heyting algebra, then so is the
otherandbothareisomorphicasskewHeytingalgebras.
Proposition 3.4. The relation D defined in (2.1) is a congruence on any skew Heyting
algebra.
K.Cvetko-Vah:OnskewHeytingalgebras 43
Proof. Let (S;∧,∨,→,1) be a skew Heyting algebra. Since D is a congruence for co-
stronglydistributiveskewlatticeswithatop, weonlyneed toprove(a → b)D(c → d)
holdsforanya,b,c,d ∈ S satisfyingaDcandbDd. Withoutlossofgeneralitywemay
assumeb≤aandd≤c. (Otherwisereplaceabyb∨a∨bandcbyd∨c∨d.)
To begin, define a map ϕ : b↑ → d↑ by setting ϕ(x) = d∨x∨d. We claim that ϕ
isalatticeisomorphismof(b↑;∧,∨)with(d↑;∧,∨),withinverseψ : d↑ → b↑givenby
ψ(y) = b∨y∨b. Itiseasilyseenthatϕandψ areinversesofeachother. Forinstance,
ψ(ϕ(x)) = b∨d∨x∨d∨bequals(b∨d∨b)∨x∨(b∨d∨b)bytheregularityof∨.
Butthelatterreducestob∨x∨bbecausebDd,andsincex∈b↑itreducesfurthertoxby
Lemma2.1,givingψ(ϕ(x))=x. ϕmustpreserve∧and∨. Indeeddistributivitygives:
ϕ(x∧x(cid:48))=d∨(x∧x(cid:48))∨d=(d∨x∨d)∧(d∨x(cid:48)∨d)=ϕ(x)∧ϕ(x(cid:48)).
Andtheregularitygives:
ϕ(x∨x(cid:48))=d∨(x∨x(cid:48))∨d=(d∨x∨d)∨(d∨x(cid:48)∨d)=ϕ(x)∨ϕ(x(cid:48)).
Thusϕ(andψ)isalatticeisomorphismofb↑withd↑. Butthenϕandψ arealsoisomor-
phismsofHeytingalgebras. Thatis,e.g.,ϕ(x→y)=ϕ(x)→ϕ(y).
Next,observethatxDϕ(x)forallx∈b↑. Indeed,regularitygives:
ϕ(x)∨x∨ϕ(x)=(d∨x∨d)∨x∨(d∨x∨d)=d∨x∨d=ϕ(x)
and likewise x ∨ ϕ(x) ∨ x = ψ(ϕ(x)) ∨ ϕ(x) ∨ ψ(ϕ(x)) = ψ(ϕ(x)) = x. There is
more: a is the unique element in its D-class belonging to b↑ and c is the unique element
inthesameD-classbelongingtod↑(sinceeachupsetu↑intersectsanyD-classinatmost
one element). But ϕ(a) in d↑ behaves in the manner just like c, and so ϕ(a) = c. Since
bD d, ϕ(b) = d∨b∨d = d and ϕ(a → b) = ϕ(a) → ϕ(b) = c → d, thus giving
a→bDc→d.
Following[5]acommutativesubsetofasymmetricskewlatticeisanon-emptysubset
whoseelementsbothjoinandmeetcommute.
Theorem3.5. Givenaco-stronglydistributiveskewlattice(S;∧,∨,1)withtop1,anop-
eration→canbedefinedonSmaking(S;∧,∨,→,1)askewHeytingalgebraifandonlyif
theoperation→canbedefinedonS/Dmaking(S/D;∧,∨,→,D )ageneralizedHeyting
1
algebra.
Proof. Tobegin,foranyuinS,theupsetu↑isaD-classtransversaloftheprincipalfilter
S ∨ u ∨ S. Secondly, the induced homomorphism ϕ : S → S/D is bijective on any
commutativesubsetofS sincedistinctcommutingelementsofS lieindistinctD-classes.
ItfollowsthatforeachuinS,ϕrestrictstoanisomorphismofupsets,ϕ : u↑ ∼= ϕ(u)↑.
u
Thuseachupsetu↑inSformsaHeytingalgebraifandonlyifeachupsetv↑inS/D,being
someϕ(u)↑,mustformaHeytingalgebra. Thetheoremfollows.
Comment. Thisresultisanear-dualoftheimportantfactthatastronglydistributiveskew
latticeSwithbottom0isthe(necessarilyunique)reductofaskewBooleanalgebraifand
only if its lattice image S/D is the reduct of a (necessarily unique) generalized Boolean
algebra. ([15],3.8.)
44 ArsMath.Contemp.12(2017)37–50
Wenextconsiderconsequencesoftheabovetheorem. Thefirstisonthe”skewlattice
side”ofthingsandthenextismoreonthe”Heytingside”. Butfirstrecallthedefinitionsof
Green’srelationsLandRonaskewlattice:
xLy ⇔(x∧y =x&y∧x=y, orequivalentlyx∨y =y&y∨x=x),
xRy ⇔(x∧y =y&y∧x=x, orequivalentlyx∨y =x&y∨x=y).
Relations L and R are contained in the Green’s relation D defined above and L◦R =
R◦L=Dholds. Askewlatticeiscalledright-handediftherelationListrivial,inwhich
caseD = R,anditiscalledleft-handediftherelationRistrivial,inwhichcaseD = L.
By Leech’s Second Decomposition Theorem [13] the relations L and R are congruences
onanyskewlatticeS,S/Risleft-handed,S/Lisright-handedandthefollowingdiagram
isapullback:
(cid:47)(cid:47)
S S/R
(cid:15)(cid:15) (cid:15)(cid:15)
(cid:47)(cid:47)
S/L S/D
Corollary 3.6. If S = (S;∧,∨,1) be a co-strongly distributive skew lattice with top 1,
thenthefollowingareequivalent:
1. SisthereductofaskewHeytingalgebra(S;∧,∨,→,1).
2. S/ListhereductofaskewHeytingalgebra(S/L;∧,∨,→,1).
3. S/RisthereductofaskewHeytingalgebra(S/R;∧,∨,→,1).
Moreover,bothLandRarecongruencesonanyskewHeytingalgebra.
Proof. Theequivalenceof(i)–(iii)isduetotheprecedingtheoremplusthefactthatS/D,
(S/L)/D and(S/R)/D areisomorphic. Next,theinducedmapρ : S → S/Lis
S/L S/R
atleastahomomorphismofco-stronglydistributiveskewlattices. Bytheargumentofthe
precedingtheorem,itinducesisomorphismsbetweencorrespondingpairsofupsets,u↑in
SandL ↑inS/L. Thusgivenx→y =(y∨x∨y)→ yandu→v =(v∨u∨v)→ v
u y v
withx;,L;,uandy;,L;,vinS,both(y∨x∨y)→ yand(v∨u∨v)→ varemapped
y v
to the common L → L , making x → y;,L;,y → v in S. A similar argument
y∨x∨y Ly y
appliestotheinducedmapλ:S→S/R.
AnalternativetothecharacterizationofTheorem3.2isgivenby:
Corollary3.7. EveryskewHeytingalgebrasatisfies:
(SHA) x(cid:22)y →zifandonlyifx∧y (cid:22)z.
Inparticular,x→y =1iffx(cid:22)y.
Ingeneral,analgebraS=(S;∧,∨,→,1)oftype(2,2,2,0)isaskewHeytingalgebra
ifthefollowingconditionshold:
1. Thereduct(S;∧,∨,1)isaco-stronglydistributiveskewlatticewithtop1.
2. y ≤x→yholdsforallx,y ∈S.
K.Cvetko-Vah:OnskewHeytingalgebras 45
3. Ssatisfiesaxiom(SHA).
Proof. GiventhatS isaskewHeytingalgebra, sincetheinducedepimorphismϕ : S →
S/DisahomomorphismofskewHeytingalgebraswehave
x(cid:22)y →ziffϕ(x)≤ϕ(y)→ϕ(z)iffϕ(x)∧ϕ(y)≤ϕ(z)iffx∧y (cid:22)z.
Conversely, let S = (S;∧,∨,→,1) be an algebra of type (2,2,2,0) satisfying (1)–(3).
Supposethatx,y,z lieinacommonupsetu ↑. Since(cid:22)isjust≤inu ↑nady → z lies
in u ↑ by (2) we have x ≤ y → z iff x ∧ y ≤ z in u ↑. (S;∧,∨,1) is thus at least
a skew Heyting lattice. Now consider the derived implication →∗ given by x →∗ y =
(y∨x∨y)→ y. Bothy →z andy →∗ z satisfy(SHA)andthusareD-equivalent. But
y
sincebothlieinthesublatticez↑,theymustbeequal.
We have seen that each skew Heyting algebra is basically a co-strongly distributive
skewlatticeSwithtop, say1, forwhichS/D isageneralizedHeytingalgebra, inwhich
case the Heyting structure on each upset u↑ of S is obtained from that of the isomorphic
upsetD ↑inS/D. ThissuggeststhatallstandardclassesofgeneralizedHeytingalgebras
u
yieldclassesofskewHeytingalgebraswhosemaximalcommutativeimagesbelongtothe
particularclass. Weconsiderseveralcases.
Case1. Finitedistributivelatticespossessawell-definedHeytingalgebrastructure. Thus
anyfiniteco-stronglydistributiveskewlatticewithatop,ormoregenerallyanyco-strongly
distributive skew lattice with a top and a finite maximal lattice image is the reduct of a
uniqueskewHeytingalgebra.
Case2. Allchainspossessingatop1formHeytingalgebras. Herethingsaresimple:
(cid:26)
1; ifx≤y.
x→y =
y; otherwise.
Thusallco-stronglydistributiveskewchainswithatopareskewHeytingalgebrareducts,
whereaskewchainisanyskewlatticeSwhereS/D isachain,i.e.,(cid:22)isatotalpreorder
onS. Here,givenx,yinacommonu↑onehas:
(cid:26)
1; ifx(cid:22)y.
x→y =
y; otherwise.
Case3. DualgeneralizedBooleanalgebras. ThesearealgebrasS=(S;∧,∨,\\,1)where
(S;∧,∨,1) is a distributive lattice with top 1 and \\ is a binary operation on S such that
(y∨x)∨(y\\x)=1and(y∨x)∧(y\\x)=yforallx,yinS. Aswith\forgeneralized
Boolean algebras, \\ is uniquely determined. Moreover, in this case, x → y = y \\x.
Adual-skewBooleanalgebraS = (S;∧,∨,\\,1)isanalgebrasuchthat(S;∧,∨,1)isa
co-stronglydistributiveskewlatticewithtop1andbinaryoperation\\suchthat:
(y∨x∨y)∨(y\\x)=1=(y\\x)∨(y∨x∨y);
(y∨x∨y)∧(y\\x)=y =(y\\x)∧(y∨x∨y).
46 ArsMath.Contemp.12(2017)37–50
Therelevantdiagramis:
1
y∨x∨y y\\x
y
Thesedualalgebrasare,ofcourse,preciselytheco-stronglydistributiveskewlatticeswith
a top whose maximal lattice images are the lattice reducts of dual-generalized Boolean
algebras. Once again we follow the commutative case: x → y = y \\x which now is
y\\(y∨x∨y)iny↑.
Wethushave:
Corollary 3.8. A co-strongly distributive skew lattice with a top S = (S;∧,∨,1) is the
reduct of a uniquely determined skew Heyting algebra (S;∧,∨,\\,1) if any one of the
followingconditionsholds:
1. S/Disfinite.
2. Sisaskewchain.
3. SisthereductofadualgeneralizedBooleanalgebra,S=(S;∧,∨,\\,1).
ImplicitinCase3isabasicdualitythatoccursforskewlattices. Givenaskewlattice
S = (S;∧,∨), its (vertical) dual is the skew lattice S(cid:108) = (S;∧(cid:108),∨(cid:108)), where as binary
functions, ∧(cid:108) = ∨ and ∨(cid:108) = ∧. Clearly S(cid:108)(cid:108) = S and any homomorphism f : S → T
of skew lattices ia also a homomorphism from S(cid:108) to T(cid:108); moreover a skew lattice S is
distributive(orsymmetric)iffS(cid:108)isthus. EitherSorS(cid:108)isstronglydistributiveifftheother
is co-strongly distributive; more generally, S or S(cid:108) is normal iff the other is co-normal.
Also,onehasabottomelementifftheotherhasatopelement,bothbeingthesameelement
inS.
Expanding the signature, (S;∧,∨,\,0) is a skew Boolean algebra if and only if its
dual (S;∧(cid:108),∨(cid:108),\\,1) is a dual skew Boolean algebra where \ and 0 are replaced by \\
and 1. Thus any skew Boolean algebra (S;∧,∨,\,0) induces a skew Heyting algebra
(S;∧(cid:108),∨(cid:108),→,1) where x → y = y \ x and 1 = old0. Standard examples of skew
Booleanalgebrasthusgiveus:
Example 3.9. Given sets X and Y, the skew Heyting operations derived from the skew
BooleanoperationsonthesetP(X,Y)ofallpartialfunctionsfromX toY areasfollows.
skewHeytingoperation description skewBooleanoperation
f ∧g f ∪(g| ) f ∨g
domg−domf
f ∨g g| f ∧g
domg∩domf
f →g g| g\f
domg−domf
1 ∅ 0