Type-2 Fuzzy Sets and BichainsJohn Harding, Carol Walker, and Elbert Walker

Department of Mathematical SciencesNew Mexico State UniversityLas Cruces, NM 88003, USA

Email: jharding@nmsu.edu, hardy@nmsu.edu, elbert@nmsu.edu

Abstract—This paper is a continuation of the study of thevariety generated by the truth value algebra of type-2 fuzzy sets.That variety and some of its reducts were shown to be generatedby finite algebras, and in particular to be locally finite. A basicquestion remaining is whether or not these algebras have finiteequational bases, and that is our principal concern in this paper.The variety generated by the truth value algebra of type-2 fuzzysets with only its two semilattice operations in its type is generatedby a four element algebra that is a bichain. Our initial goal is tounderstand the equational properties of this particular bichain,and in particular whether or not it has a finite equational basis.

I. INTRODUCTION

The underlying set of the algebra of truth values of type-2fuzzy sets is the set M = Map([0, 1], [0, 1]) of all functionsfrom the unit interval into itself. This set is equipped withthe binary operations � and �, the unary operation ∗, and thenullary operations 1̄ and 0̄ as spelled out below, where ∨ and∧ denote maximum and minimum, respectively.

(f � g) (x) = sup {f(y) ∧ g(z) : y ∨ z = x}(f � g) (x) = sup {f(y) ∧ g(z) : y ∧ z = x}

f∗(x) = sup {f(y) : 1 − y = x} = f(1 − x)

1̄(x) ={

0 if x �= 11 if x = 1 0̄(x) =

{1 if x = 00 if x �= 0

The algebra of truth values of type-2 fuzzy sets was introducedby Zadeh in 1975, generalizing the truth value algebras ofordinary fuzzy sets, and of interval-valued fuzzy sets. Thedefinitions of the convolutions �, �, and ∗ are sometimesreferred to as Zadeh’s extension principle.

Definition 1: The algebra M = (M,�,�,∗ , 1̄, 0̄) is thealgebra of truth values for fuzzy sets of type-2.

Type-2 fuzzy sets, that is, fuzzy sets with this algebraM of truth values, play an increasingly important role inapplications, making M of some theoretical interest. See, forexample, [3], [8], [11], [12], [13], [18].

We are concerned here with the equational properties ofthis algebra, much as one is concerned with the equationalproperties of the Boolean algebras used in classical logic. Themain question we are interested in is whether there is a finiteequational basis for the variety V (M) generated by M. Wehave made some progress toward this, and other questions,but it remains open.

An important step in understanding the equational theoryof M was taken in [4], [17] where the operations �,� were

written in a tractable way using the auxiliary operationsL, R where fL, fR are the least increasing and decreasingfunctions, respectively, above f . Using this, it was shown thatM satisfies the following equations.

Proposition 2: Let f, g, h ∈ M .

1) f � f = f ; f � f = f2) f � g = g � f ; f � g = g � f3) f � (g � h) = (f � g) � h; f � (g � h) = (f � g) � h4) f � (f � g) = f � (f � g)5) 1̄ � f = f ; 0̄ � f = f6) f∗∗ = f7) (f � g)∗ = f∗ � g∗; (f � g)∗ = f∗ � g∗

Algebras, such as M, that satisfy the above equations, exceptpossibly 4), have been studied in the literature under the nameDe Morgan bisemilattices [1], [9], [10].

Definition 3: A variety of algebras is the class of allalgebras of a given type satisfying a given set of identities (abasis for the variety). Equivalently (by a famous theorem ofBirkhoff), a variety is a class of algebras of the same typewhich is closed under the taking of homomorphic images,subalgebras and (direct) products.

Definition 4: For an algebra A, the variety V(A) generatedby A is the class of all algebras with the same type as A thatsatisfy the same equations as A. An algebra A is locally finiteif each finite subset of A generates a finite subalgebra of A,and a variety is locally finite if each algebra in the variety islocally finite.

An advance in understanding M and its equational proper-ties came in [7], where it was shown that the variety V(M) isgenerated by a 12-element De Morgan bisemilattice. It followsthat V(M) is locally finite, and that there is an algorithm todetermine whether an equation holds in M.

It is natural to consider whether the equations in Proposition2 could be a basis for the variety V(M). This is not the case asV(M) is locally finite, and there are De Morgan bisemilatticesthat are not locally finite, such as certain ortholattices. So tofind a basis for the variety V(M) one must add equations tothis list. Whether there is a finite basis for V(M) remains open.

The observant reader at this point will have consideredBaker’s Theorem [2], that says a finitely generated congruencedistributive variety has a finite basis. Unfortunately we cannotapply this result as V(M) is not congruence distributive, as isnoted in a later section.

978-1-4244-7858-6/10/$26.00 ©2010 IEEE

We decided to simplify the problem somewhat, and restrictour attention to equations involving only the operations �,�and not using the negation ∗ or constants 1̄, 0̄. Of course thereduct (M,�,�) of M to this type satisfies equations 1) - 4)above.

Definition 5: An algebra (A,�,�) with two binary opera-tions is called a bisemilattice if it satisfies equations 1) - 3)above, and a Birkhoff bisemilattice if it satisfies equations 1)- 4) above.

So our reduct (M,�,�) is a Birkhoff bisemilattice. Ofcourse, the variety generated by this algebra is generated bythe reduct of the 12-element De Morgan system that generatesV(M), but here one can do much better. In [7] it was shownthat the variety generated by (M,�,�) is generated by the4-element bisemilattice we call B, shown in Figure 1.

�

�

�

�

�

�

�

�

� �

1

2

3

4

1

3

2

4

Fig. 1. The 4-element bichain B

In any bisemilattice, the operations �,� induce two partialorderings. In Figure 1, the partial ordering for � is shownat left, that for � at right. One understands � as meet in itspartial ordering, and � as join in its partial ordering. The twopartial orderings coincide if, and only if, the bisemilattice is alattice. The two partial orderings in this figure are chains, andfor this reason the bisemilattice B is called a bichain.

While there is a considerable literature on bisemilattices(see, for example, [10], [15], [16]), there seems to be relativelylittle known about the quite natural case of bichains. Ourefforts here are largely devoted to studying bichains and thevarieties they generate. We believe this is of interest for its ownsake, as well as for its application to understanding equationalproperties of M.

II. PRELIMINARIES

Let BiSemi be the variety of all bisemilattices, Birkbe the variety of all Birkhoff systems, BiCh be the varietygenerated by all bichains, DL be the variety of all distributivelattices, and SL be the variety of all bisemilattices satisfyingx� y = x� y, which is called the variety of semilattices. Forany bisemilattice S we let V(S) be the variety generated by S.To ease notation, we often write equations using + in place of�, using juxtaposition for �, and binding juxtaposition moretightly than addition.

Proposition 6: Every bichain is a Birkhoff system, soBiCh⊆Birk.

Proof: Suppose x, y are elements of a bichain. Then eachof xy and x+y is either x or y, and we check that in the fourpossible cases x(x + y) = x + xy.

We will work not only with the bichain B, but with otherbichains as well. When describing these, we assume the �-order is 1 < 2 < · · · < n and then just give the �-order.Any ordering of 1, 2, .., n for the �-order gives a bichain, soup to isomorphism there are n! n-element bichains. Below wedescribe and name all bichains with two or three elements.

�

�

D1

1

2

�

�

D2

2

1

�

�

�

A1

1

2

3

�

�

�

A2

1

3

2

�

�

�

A3

2

1

3

�

�

�

A4

2

3

1

�

�

�

A5

3

1

2

�

�

�

A6

3

2

1

Fig. 2. The 2- and 3-element bichains

Note that D1 and A1 are distributive lattices so generatethe variety DL, and D2 and A6 are semilattices so generateSL [2]. By [14] the join of DL and SL is the varietyof distributive bisemilattices; that is, bisemilattices satisfyingboth distributive laws. As D1 and D2 are subalgebras of A4,and A4 is a quotient of their product, A4 generates DL∨SL.By [10] the variety of bisemilattices satisfying the meet-distributive law x(y + z) = xy + xz covers the distributivebisemilattices, as does the variety of bisemilattices satisfyingthe join-distributive law x + yz = (x + y)(x + z). As A2

satisfies meet-distributivity but not join distributivity, and A3

satisfies join distributivity but not meet distributivity, V(A2)and V(A3) cover V(A4). As A2 and A3 are subalgebras ofB, we have V(A2) ∨ V(A3) is contained in V(B). Using theUniversal Algebra calculator [5] we can find an equation toshow this containment is strict. This program also providesequations to show neither V(A2) nor V(A3) are contained inV(A5), and V(A5) is not contained in V(B).

A diagram of the containments between these varieties isgiven below.

�

� �

�

� �

�

�

�

Trivial

���

���

��

��

�

��

��

��

���

��

��������

V(A1) V(A6)

V(A4)

V(A2) V(A3)

V(B)

V(A5)

Fig. 3. Containments between varieties

Our conjecture is that V(B) is the largest subvariety ofBiCh not containing A5, a situation known as a splitting.We also have a candidate for a splitting equation:

[x(y + z)][xy + xz] = [x(y + z)] + [xy + xz] (S)

If this is indeed the case, V(B) is defined by the singleequation (S) together with equations defining BiCh. We havenot carefully considered yet whether the variety BiCh isfinitely based.

III. BICHAINS IN THE VARIETY V(B)

In this section we outline a proof of the following.

Theorem 7: For a bichain C, the following are equivalent.

1) C ∈ V(B).2) A5 is not a subalgebra of C.3) C satisfies (S).

Proof: (1 ⇒ 3) This is of course is simply a matter ofchecking the equation (S) holds in B, but the situation is abit more interesting than this. Note there is a congruence onB that collapses only the two middle elements {2, 3}, and theresulting quotient is a distributive lattice. Take any equations = t that holds in all distributive lattices. If this equation isto fail in B for some choice of elements, it must be that s, tevaluate to 2 and 3. As {2, 3} is a subalgebra of B isomorphicto the 2-element semilattice, it then follows that st = s + tholds in B. The equation (S) is an instance of this, taking s = tto be the meet distributive law.

(3 ⇒ 2) Take x = 2, y = 1, z = 3 to see A5 does notsatisfy (S).

(2 ⇒ 1) To show C ∈ V(B) it is sufficient to show everyfinite sub-bichain of C belongs to V(B). Indeed, if C �∈ V(B),there is some equation valid in B that fails in C. This equationinvolves only finitely many variables, so there is some finitelygenerated subalgebra of C that does not belong to V(B). Butas C is a bichain, every subset of C is in fact a subalgebra ofC. So to show 2 ⇒ 1, it is enough to show this for C a finitebichain.

We show by induction on n = |C| that A5 �≤ C impliesC ∈ V(B). For n ≤ 3 all n-element bichains are given inthe figure in the previous section, and all but A5 are shownto belong to V(B). Suppose C has n ≥ 4 elements. We firstestablish a lemma that handles several cases.

Lemma 8: For a finite bichain C, let C∪{∞} be the bichainformed from C by adding a new element to the bottom of the�-order and the top of the �-order; let C∪{b} be formed fromC by adding a new element to the bottom of both orders; andlet C ∪ {t} be formed from C by adding a new element tothe top of both orders. Then if C ∈ V(B), so are C ∪ {∞},C ∪ {b}, and C ∪ {t}.

Proof of Lemma: We first show B ∪ {∞}, B ∪ {b} andB ∪ {t} belong to V(B). Note B ∪ {∞} is the quotient ofB × D2 by the congruence θ that has one non-trivial blockconsisting of B × {1}; B ∪ {b} is the subalgebra of B × D1

consisting of B×{2} and (1, 1); and B∪{t} is the subalgebra

of B × D2 consisting of B × {1} and (4, 2). As D1 and D2

belong to V(B), so do these algebras.Assume C belongs to V(B). Then there is a set I , a

subalgebra S ≤ BI , and an onto homomorphism ϕ : S → C.

Consider the constant function ∞ in (B ∪ {∞})I whoseconstant value is the new element ∞ added to B. In B,x � ∞ = ∞ and x � ∞ = ∞. It follows that S ∪ {∞}is asubalgebra of this power, and ϕ extends to a homomorphismfrom S∪{∞} onto C∪{∞}. The arguments for C∪{b} andC ∪ {t} are similar, using powers of B ∪ {b} and B ∪ {t}.

Proof of Theorem continued: Assume the �-order of C

is 1 < 2 < · · · < n. If the bottom element of the �-orderof C is 1, then C is isomorphic to C

′ ∪ {b} where C′ is the

sub-bichain {2, . . . , n} of C. Then by the inductive hypothesisand the above lemma, C ∈ V(B). A similar argument handlesthe cases where either 1 or n is the top element of the �-orderof C. Set

U = {k : 2 ≤ k ≤ n and k precedes 1 in the � -order}V = {k : 2 ≤ k ≤ n and 1 precedes k in the � -order}

As 1 is not the bottom or top of the �-order, U and V arenon-empty. Also, as A5 is not a subalgebra of C, if u ∈ U andv ∈ V , then u < v. Also, as n is not the top element of the�-order, V must have at least two elements. So there is some2 ≤ k ≤ n− 2 with U = {2, . . . , k} and V = {k +1, . . . , n}.

There are congruences θ, φ on C with θ collapsing{1, . . . , k} and nothing else, and φ collapsing V and nothingelse. Note C/θ is isomorphic to the sub-bichain {1, k +1, . . . , n} of C, and C/φ is isomorphic to the sub-bichain{1, . . . , k, k+1} of C. It follows from the inductive hypothesisthat C/θ and C/φ belong to V(B). As θ and φ intersect tothe diagonal, C is a subalgebra of their product, so belongs toV(B).

Remark 9: At this point, if we had congruence distribu-tivity, it would follow that every subdirectly irreducible inthe variety BiCh is a bichain, and then the above theoremwould imply V(B) is defined, relative to the equations definingBiCh, by the single equation (S). However we do not havecongruence distributivity [15].

IV. SPLITTING

In this section we investigate projectivity and splitting forvarious bichains, and in particular for A5. Our main result hereshows there is a largest subvariety of BiCh not containingA5, and the results of the previous section lead us to believethis may be the variety V(B).

Definition 10: An algebra P is weakly projective in avariety V if for every homomorphism f : P → E and everyonto homomorphism g : A � E, there is a homomorphismh : P → A with gh = f .

Note: The usual definition of projective uses an epimor-phism in place of the onto homomorphism g. In a variety V ,there may be more epimorphisms than onto homomorphisms,so an algebra that is weakly projective may not be projective.

Of course, the definition of weakly projective in V requiresall algebras to come from the variety V .

The following well-known result [6] is a convenient refor-mulation.

Proposition 11: P is weakly projective in V if and onlyif for every onto homomorphism u : A � P, there is anembedding r : P → A with u ◦ r = idP.

Weak projectives are of interest for several reasons, but ourprimary one lies in the following. First, for an algebra P in avariety V define

W(P) = {A ∈ V : P �↪→ A}Here P �↪→ A means P is not isomorphic to a subalgebra of A.

Proposition 12: If P is weakly projective in V and subdi-rectly irreducible, then W(P) is a variety, and is the largestsubvariety of V that does not contain P.

This is a well-known result [6] and not difficult to prove.The situation is sometimes referred to as a splitting, as itsplits the lattice of subvarieties of V into two parts, thosethat contain the variety V(P), and those that are containedin W(P). Further, such a splitting comes equipped with anequation, called the splitting equation, defining the varietyW(P) relative to the equations defining V , although the exactform of this equation may be difficult to establish. We nowapply these results in our setting.

Proposition 13: The 2-element distributive lattice D1 issubdirectly irreducible and weakly projective in BiCh. Itssplitting variety W(D1) is the variety SL of semilattices.

Proof: Clearly D1 is subdirectly irreducible (see Figure2). Let A be in BiCh and f : A � D1 be an ontohomomorphism. Then there are x, y in A with f(x) = 1 andf(y) = 2. Then f(xy) = 1 and f(x+y) = 2, so xy is differentfrom x + y. In any Birkhoff system we have xy(x + y) =xyx(x + y) = xy(x + xy) = xy(xy + x) = xy + xyx =xy + xy = xy and similarly, x + y + xy = x + y. So thereis a homomorphism r : D1 → A defined by r(1) = xy andr(2) = x + y and f ◦ r = idD. Thus D1 is weakly projective.

To see W(D1) = SL, note that the two-element semilatticeD2 belongs to W(D1), so one containment is trivial. For theother, suppose A does not belong to SL. Then there are x, y ∈A with xy not equal to x+y, giving {xy, x+y} is a subalgebraof A isomorphic to D1, so A �∈ W(D1).

Note: For D1, these results hold also in the larger varietyBirk.

Proposition 14: The 2-element semilattice D2 is subdirectlyirreducible and weakly projective in BiCh. Its splittingvariety W(D2) is the variety DL of distributive lattices.

Proof: Clearly D2 is subdirectly irreducible (see Figure2). Let A be in BiCh and f : A � D2 be an ontohomomorphism. Then there are x, y ∈ A with f(x) = 1 andf(y) = 2. While we could now just jump to the answer, let’sbuild it step at a time to see the idea for later. This same

idea would have worked above. We first patch up the meetoperation and consider the following:

y xyxy y

� �Fig. 4. Fixing meets

We have f(xy) = f(x)f(y) = (1) (2) = 1 and f(y) = 2.Also (xy)y = xy. So {x, xy} is a 2-element subset of A thatworks well with respect to meet. But it doesn’t work well withrespect to join since we would like that y+xy = xy and thereis no reason for this to be true. We work on what we havenow and get it to work well with respect to join.

y y + xyy + xy y

� �Fig. 5. Fixing joins

Now by construction, this works well with respect to join, asy + y + xy = y + xy. It also works with respect to meet, asy + xy = y(x + y), so y(y + xy) = y + xy. So {y + xy, y}is a subalgebra of A, f(y + xy) = 2 + 1 = 1 and f(y) = 2.So there is r : D2 → A with r(1) = y + xy and r(2) = y, soD2 is weakly projective.

We next show W(D2) = DL. Surely W(D2) ⊇ DL. Toshow W(D2) ⊆ DL, suppose A ∈ BiCh and A has nosubalgebra isomorphic to D2. Note that for any x, y ∈ A wehave

x[x(x + y)] = x(x + y),

and Birkhoff’s equation a(a + b) = a + ab gives

x + x(x + y) = x(x + x + y) = x (x + y) .

As A has no subalgebra isomorphic to D2 it follows that x =x(x + y) for each x, y ∈ A, and then by Birkhoff’s equationthat x = x + xy for each x, y ∈ A. So A is a lattice.

Consider the equations

1) x(x + y)(xz + y) = x(x + y)(xz + y + z)2) z(x + y)(y + xz) = z(x + y)(y + z + xz)Both hold in every bichain. To see this, as these equations

involve three variables it is enough to check them in each 3-element bichain, and this is not difficult. So these equationshold in the variety BiCh, hence also in A. The first does nothold in the 5-element modular, non-distributive lattice M3,and the second does not hold in the 5-element non-modularlattice N5. So A is a lattice containing neither M3 nor N5 asa subalgebra, showing A is a distributive lattice [2].

Note: Our proof shows more. The algebra D2 is weakly pro-jective in the larger variety Birk. It therefore has a splittingvariety in Birk, but this is not DL, but the variety Lat ofall lattices. This proof also shows Lat ∩ BiCh = DL.

Now to the result most pertinent to our variety V (B). Forconvenience, we recall what A5 looks like.

3 22 11 3

� �Fig. 6. The bichain A5

Proposition 15: A5 is subdirectly irreducible and weaklyprojective in BiCh, and its splitting variety W(A5) containsV(B).

Proof: The bichain A5 is subdirectly irreducible with itsminimal congruence being the one collapsing 1 and 2. To seeit is weakly projective, assume A ∈ BiCh and f : A � A5.Then there are x, y, z in A with f(x) = 1, f(y) = 2, f(z) = 3.We follow the process in the previous proof to try to build asubalgebra of A that is isomorphic to A5. As our first step,we try to fix up meets.

z yzyz xyzxyz z

� �Fig. 7. Fixing meets

So now meets are okay, but joins are a problem. We try to fixthem, bearing in mind we may wreck our meets when we do.

z z + xyz + yzz + xyz + yz z + xyz

z + xyz z

� �Fig. 8. Fixing joins

So now we have fixed up joins, but have troubles with themeets again. Before we continue further, let’s look at a coupleof equations that hold in BiCh:

1) z + xyz = z(z + xy)2) z + xyz + yz = z(z + xy) + z(z + y)3) a(ab + ac) = ab + ac

The first follows from Birkhoff’s equation, the second byBirkhoff’s equation after including an extra + z, and the thirdcan be seen to hold in all 3-element bichains, hence in BiCh.

Consider meets again making use of these equations. Weget

z(z + xyz) = zz(z + xy) = z(z + xy) = z + xyz.

So the meet of the top and bottom at left works as it should.Also,

z(z + xyz + yz) = z(z(z + xy) + z(z + y))= z(z + xy) + z(z + y)= z + xyz + yz.

So the meet of the top and the middle elements at left worksas it should. But the meet of the middle and the bottom at leftposes a problem. So we go one more round, and patch thingsup again with the meets. This time we only have to fix themeet of the bottom two, and of course, make correspondingchange to the element in the middle on the right side.

z z + xyz + yzz + xyz + yz (z + xyz)(z + xyz + yz)

(z + xyz)(z + xyz + yz) z

� �Fig. 9. Fixing meets again

So now we have the meets on the left work as they should,we need to check the joins on the right. We haven’t changedthe bottom and top, so their join is as it should be. We usethe dual of the third equation above, a + (a + b)(a + c) =(a + b)(a + c), which will also hold in all bichains, hence inBiCh. Apply this to the join of the bottom and middle onthe right to get

z + (z + xyz)(z + xyz + yz) = (z + xyz)(z + xyz + yz)

which is as it should be. We have still to check the join of themiddle and top on the right side. Letting u = z + xyz + yzand v = z + xyz, and noting u + v = u, we get

z + xyz + yz + (z + xyz + yz)(z + xyz) = u + uv

= u(u + v)= uu

= u

= z + xyz + yz.

So the join of the middle and top on the right side is also asit should be, and everything works fine. We then get that

{z, z + xyz + yz, (z + xyz)(z + xyz + yz)}is a subalgebra of A. One easily sees that

f((z + xyz)(z + xyz + yz)) = 1f(z + xyz + yz) = 2

f(z) = 3.

So A5 is weakly projective. That W(A5) contains V(B) istrivial as A5 is not a subalgebra of B.

V. CONCLUSIONS AND REMARKS

From a previous paper [7], we know that the varietygenerated by the truth value algebra of type-2 fuzzy sets withonly its two semilattice operations in its type is generated bya 4-element algebra B that is a bichain and, in particular, aBirkhoff system.

Our aim is to find an equational basis for the varietygenerated by B. This problem seems difficult, but we havesome progress. Our technique is to consider a particular 3-element bichain A5, show it is subdirectly irreducible and

weakly projective, hence splitting, and that its splitting varietyW(A5) in BiCh contains V(B).

We conjecture that W(A5) = V(B). If so, this will showthat the splitting equation (S) for A5 then defines V(B) withinBiCh. The results of the previous section lend credence tothis as we have shown a bichain belongs to W(A5) if andonly if it belongs to V(B). But this remains an open problem.

There remain a number of open problems in connection withthis work. These include determining whether W(A5) = V(B)and finding an equational basis for BiCh. Together, these willprovide an equational basis for V(B), and hence for the �,�fragment of the truth value algebra M of type-2 fuzzy sets.

Of independent interest is the matter of understanding whichbichains are weakly projective, and determining the lattice ofsubvarieties of BiCh.

Remark 16: In preparing this still incomplete work, wemade extensive use of the Universal Algebra Calculator, aswell as the programs Prover9 and Mace4 to find and workwith equations. After finding equations with these programswe further verified all properties by hand. We are grateful toseveral people for providing equations of help to us, includingPeter Jipsen, Keith Kearnes, and Fred Linton, and also to AnaRomanowska for several communications.

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