Free algebras over a poset in varieties generated by a finite number of algebras

Free algebras over a poset in varieties generated by a finite number of algebras

Aldo Figallo, Jr.

March 26, 2009

Abstract

In 1945, the notion of free lattice over a poset was introduced by R. Dilworth (Trans. Am. Math. Soc. 57, (1945) 123–154). Later, in Monteiro’s school, constructions of free algeras over a poset for differ- ent classes of algebras were presented. In this paper, a construction for this free algebra is exhibit in the case that the class of algebras is a variety generated by a finite number of algebras. On the other hand, neccesary and sufficient conditions for the existence of such free alge- bra are established. Finally, this construction is applied in different classes of algebras.

Keywords. Free algebras, Free algebras over a poset, varieties of alge- bras, algebras related to logic

AMS subject classification (2000). 08B20, 03G10, 03C05, 03G25

1 Introduction

In 1945, R. Dilworth ([4]) introduced the notion of free lattice over a poset.

Later, this notion was studied by diferent authors. In particular, in 1967, a construction for the free distributive lattice over a poset was indicated (see [4]).

On the other hand, the free algebra over a poset in other classes of algebras were also constructed as for example in De Morgan algebras ([12], [8]), Tarski algebras ([13]), Lukasiewicz, Post and Moisil 3-valued algebras ([9]).

In the case of Tarski algebras, the structure of the free algebra over a poset as well as its cardinal has been determined for two particular cases:

when I is the antichain and when it is a finite chain.

In order to generalized what has been proved in [12] for Hilbert algebras generated by chains we have obtained a more general construction that we shall detail in the next section.

2 Free algebras over a poset, construccion

Let us consider a variety of algebras that have an underlying order structure definable by means of certain equations pie(x, y) = qi(x, y), 1 ≤ i ≤ n in terms of the operations of the algebra. This is the case of Hilbert algebras ([3]), distributive lattices, etc. Indeed,

(i) in a Hilbert algebra, i= 1, p₁(x, y) =x→y, q₁(x, y) =x→x, (ii) in a distributive lattice, i= 1 and the equations can be:

(a) p₁(x, y) =x∧y, q₁(x, y) =x, or (b) p₁(x, y) =x∨y, q₁(x, y) =y.

Then, the notion of free algebra over a posetI relative toVcan be defined as follows:

Consider a variety of algebras that have an underlying order structure definable by means of some set of equations and consider a poset I.

Definition 1. A V−algebra L_V(I) is said to be free over I if the following conditions are satisfied:

(L1) there is an order monomorphism g : I −→ L_V(I) such that L_V(I) = [g(I)]_V, where [X]_V is V-subalgebra of L_V(I) generated by X,

(L2) for each V−algebra A and each order-preserving function f : I → A, there is an homomorphism h:L_V(I)−→A such that h◦g =f.

Now, we are going to determine the structure ofL_V(I) in the particular case that the variety has a finite number of subdirectly irreducible algebras S_i, 1≤i≤n such that the order of C=

n

Q

i=1

S_i is not an antichain.

LetI be a poset and E be the set of all order-preserving functions onI in the V−algebra C. Consider the function g : I −→ C^E defined by g(i) =G_i where G_i(f) = f(i), for all f ∈ E and i ∈ I. Then, the following assertion holds.

Theorem 2. L= [G]_V is the free V−algebra overI, whereG={G_i :i∈I}.

Proof. (L1) It is clear that if i≤j, then G_i ≤G_j, for all i, j ∈I.

On the other hand, let us suppose thatG_i ≤G_j and that there arei, j ∈I such that i 6≤ j. Consider now a, b ∈ C such that a < b and the function f^∗ :I −→C given by

f^∗(k) =

b if k≥i

a in other case .

Then, f^∗ ∈E,f^∗(i) =b and f^∗(j) =a, which is a contradiction.

Therefore, g is an order monomorphism. Besides, L = [g(I)]_V since we have that G=g(I) and by construction L= [G]_V.

(L2): LetA be a V−algebra and let f : I −→ A be an order-preserving function. SinceV=V(C) =HSP(C),Ais isomorphic to a subalgebraA^∗ de C^X, whereXis an arbitrary set. Then, there is an isomorphismϕ:A−→A^∗ defined by the prescription ϕ(a) = H_a where H_a ∈C^X for each a ∈A. Let us consider also the function ϕ^∗ =ϕ◦f whereϕ^∗(i) =ϕ(f(i)) =H_f(i).

Let x0 ∈ X and αx0 : I −→ C be the function defined by αx0(i) = H_f(i)(x₀). Then, we have that α_x0 ∈E. On the other hand, let k:X −→E, defined in the following way k(x) = α_x for all x ∈ X. Then, it is simple to verify that the application h : L −→ C^X defined by h(F) = F where F(x) =F(k(x)) is an homomorphism which verifies (h◦g)(i) = ϕ^∗(i), for all i∈I.

Finally, it is shown without any difficulty thath(L)⊆A^∗.

An interesting consequence of Theorem 2’s proof, whose demonstration is immediate, is the following.

Corollary 3. IfC is antichain, the free algebra over the posetI exists if and only if I is antichain.

Besides, in the particular case that the variety is generated by a finite chain a direct consequence of Theorem 2 is the following

Corollary 4. IfE_S ={f ∈E : [f(I)] =S /C}, thenL_V(I)⊆ Q

S/C

S^ES ⊆C^E.

In what follows we shall apply the results obtained above in different varieties of algebras.

3 Applications and examples 3.1 Implication algebras

In 1967, J. Abbott ([1]), presented the implication algebras as the algebraic models the classic implicative propositional calculus. These algebras can be defined as algebras hI,→,1i of type (2,1) which satisfy the following identities:

(I1) 1→p=p, (I2) p→p= 1,

(I3) p→(q→r) = (p→q)→(p→r), (I4) (p→q)→q= (q→p)→p.

It is well–known that these algebras coincide with the variety of Hilbert algebras generated by a 2-element chain.

Consider now a finite posetI and the varietyI of all implication algebras.

Lemma 5. Let F ∈ LI(I) and D ∈ Π(2^E). Then, F ∨D ∈ LI(I), where Π(2^E) is the set of atoms of 2^E and the set E is the one consider in Theo- rem 2.

Proof. Let E = {h₁, . . . , h_s} with s < ω. IfD ∈ Π(2^E), then there is only one h_j ∈ E, 1 ≤ j ≤ s such that D(h_j) = 1 and D(h_k) = 0 for all k 6= j, with 1≤k ≤s. Then, it is verified that D= V

i∈I

Gi0

where

G⁰i =

Gi, if Gi(hj) = 1

−G_i, if G_i(h_j) = 0 .

It is clear that there isHi ∈LI(I) such thatF∨Gi0

=Hi→F, from which we have F ∨D =F ∨V

i∈I

G_i⁰ = V

i∈I

(F ∨G_i⁰) = V

i∈I

(H_i→F) = V

i∈I

(−H_i∨F) =

−(W

i∈I

H_i)∨F = (W

i∈I

H_i)→F ∈L_I(I).

On the other hand, we can see that the following lemma is verified for any I finite.

Lemma 6. L_I(I) = S

i∈I

F(G_i) where F(G_i) is the principal filter generated by G_i in 2^E.

Proof.

Since G_i ∈ F(G_i) for all i ∈ I and S

i∈I

F(G_i) is a subalgebra of 2^E, we have that L_I(I)⊆ S

i∈I

F(G_i).

Conversely, if F ∈ L, then there is i₀ ∈ I such that F ∈ F(G_i0) and therefore G_i0 ≤ F. Besides, as I is a finite set we have that 2^E is a finite Boolean algebra. Since F ∈ 2^E we can assert that F = W

{DF : DF ∈ Π(2^E), D_F ≤ F}=G_i0 ∨W{D_F : D_F ∈ Π(2^E), D_F ≤F}= W{(G_i0 ∨D_F) : D_F ∈Π(2^E), D_F ≤F}. Then, by Lemma 5, we have thatF ∈L_I(I).

In order to determine the number of elements ofL_I(I) whenI is an antichain with n elements, we have:

|L_T(I)|=|

n

[

i=1

F(G_i))|=

n

X

k=1

(−1)^k+1α_k(n),

whereαk(n) = P

1≤i₁<···<i_k≤n

|F(Gi1)∩F(Gi2)∩ · · · ∩F(Gi_k)|

Then, by the symmetry of the problem,

|

k

\

j=1

F(Gij)|=|

k

\

t=1

F(Gt)|

from which we have

|L_T(I)|=

n

X

k=1

(−1)^k+1 n

k

|

k

\

i=1

F(G_i)|.

Using any result of those established in [10, 7] for computing|

k

T

i=1

F(Gi)|

we have:

Theorem 7. |L_I(I)| =

m

P

k=1

(−1)^k+1 m

k

2^2m−k, where |L_I(I)| is the cardinal of the free implication algebra with n generators.

3.2 2-valued implicative semilattices

n-valued implicative semilattices were introduced by M. Canals Frau and A.

V. Figallo in [6] and in the case n = 2 they can be presented as algebras hA,→,∧,1i of type (2,2,0) which satisfy the following conditions:

(IS21) x→x= 1, (IS₂2) (x→y)∧y=y,

(IS₂3) x→(y∧z) = (x→z)∧(x→y), (IS₂4) x∧(x→y) =x∧y,

(IS₂5) ((x→y)→x)→x= 1.

Observe that the class these algebras constitutes a variety which we shall denote by IS2 and whose member we shall callIS2-algebras. IS2 is gener- ated by the algebrah{0,1},→,∧,1iwhere→satisfies the identities 0→x= 1 y 1→x=x.

Now, we shall study the free IS2-algebra over a poset.

Lemma 8. L_IS2(I) = [V

i∈I

G_i).

Proof. It is clear that (1) G_i ∈ [V

i∈I

G_i) for all i∈ I. Besides, given F, H ∈ [V

i∈I

G_i), since H ≤ F →H we have that F →H ∈ [V

i∈I

G_i). On the other hand, it is simple to verify that F ∧H ∈ [V

i∈I

G_i). Then, [V

i∈I

G_i) ∈ IS₂ and from (1) we conclude L_IS2(I) ⊆ [V

i∈I

G_i). The other inclusion is immediate.

Now, we shall determine the number of elements of the freeIS2−algebra overI, which we shall note L_IS2(n), whenI is an antichain withn elements.

From Lemma 8 we have thatLIS2(n) is a Boolean algebra. Then,

L_IS2(n)' Y

D∈E(LIS2(n))

L_IS2(n)/D,

whereE(L_IS2(n)) is the set of maximal deductive systems of the algebra.

Therefore,

L_IS2(n)'B₂^α,

whereα =|E(LIS2(n))|y B2 is the Boolean algebra with 2 elements.

Observe that V

i∈I

G_i 6∈ M for all M ∈ E(L_IS2(n)) since in other case it would result M =L_IS2(n) which would be a contradiction.

On the other hand, let β : Epi(L_IS2(n),B₂) −→ E(L_IS2(n)) be the application defined by β(H) = Ker(H), where Ker(H) = {f ∈ LIS2(n) : H(f) = 1}. Then, it is simple to verify that β is a bijection and therefore we may infer that

|E(LIS2(n))|=|Epi(LIS2(n),B2)|.

Leth∈Epi(L_IS2(n),B₂) and h=h/G:G→B₂. Since V

i∈I

G_i 6∈Ker(h), we have

0 =h(^

i∈I

G_i) = ^

i∈I

h(G_i) = ^

i∈I

h(G_i).

Therefore, there is i₀, 1 ≤ i₀ ≤ n such that h(G_i0) = 0. Then, if we consider the set

F ={f :G→B₂ :f(G_i) = 0 for somei,1≤i≤n}

it results that|Epi(LIS2(n),B2)|=|F| from which we deduce

α=|F|= 2ⁿ−1.

Then, it is proved that Theorem 9. |LIS2(n)|= 2²ⁿ⁻¹.

3.3 De Morgan algebras

In what follows we shall explain the construction of the free De Morgan algebra over a chain with 2 elements.

Definition 10. ([2]) An algebra L = hL,∨,∧,∼,0,1i of type (2,2,1,0,0) is a De Morgan algebra if the reduct hL,∨,∧,0,1iis a bounded distributive lattice which satisfies the following identities:

(dM1) ∼∼x=x,

(dM2) ∼(x∧y) =∼x∨ ∼y.

The next Hasse diagrams represent the poset I and the algebra K that generates the variety of De Morgan algebras:

...........................................................................................................................................................................................................................................................

.

•0

•1

•

a=∼a •b=∼b K

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

•1

•2 I

Then, the free De Morgan algebraM(I) overI has the following diagram:

...........................

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...

...

...

...

...

...

...

...

...

...

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......................................................................................................................................................

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. . .. . . .. . .. . . .. . . .. . . .. . .. . . .. . . .. . .. . . .. . . .. . . .. . .. . . .. . . ..

. .. .. .. . .. .. .. . .. .. . .. .. .. . .. .. .. . .. .. .. . .. .. .. . .. .. .. . .. .. .. . .. .. . .. .. .

.. . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . .. . . .. . . .. . .. . . .. . . .. . . .. . .. . . .. . . .. . . .. . .. . . .. . . .. . .. . . .. . . .. . . .. . .. . . .. . . .. . .. . ..

.. .. .. . .. .. . .. .. .. . .. .. .. . .. .. .. . .. .. .. . .. .. .. . .. .. . .. .. .. . .. .. .. . .. ..

............................

. . .. .. .. . .. .. .. . .. .. . .. .. .. . .. .. .. . .. .. .. . .. .. .. . .. .. .. . .. .. .. . .. .. . .. ..

...........................

...

...

...

...

...

•

•

•

•

•

•_O A

B C

D

F

G=G1

H E

J L

K M

N P

R

Q=G₂ S

T 1

M(I)

...

..

...

..

...

..

...

..

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....

...............

. .. . .. .. .. . .. .. .. . .. .. .. . .. .. .. . .. .. .. .

where A=G₁∧ ∼ G₂, B= G₂∧ ∼ G₂, C= G₁∧ ∼ G₁, F=B∨C, H=∼

G₂∨C, E=B∨G₁, J=∼ G₂∨G₁.

3.4 Distributive Lattices

Now, we shall build the free distributive lattice over a posetIwith 3 elements which is represented by the following diagram:

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

...

....

•b

•c

• a

I

...

...

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...

...

...

...

...

...

...

...

...

...

...

....

•0

•1

L

where L is the algebra which generates the variety of lattices and the free distributive lattice L(I) over the poset I has the following diagram:

...........................................................................

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.........

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..........

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•0

•1

•

•

•

•

•

•

•

• G₁

B

H F

E

G₃

G2

A

L(I)

where G₁, G₂ y G₃ are the generators of L(I). Besides, the other elements are obtained in the following way:

A=G1∧G2, B =G1∧G3, F =G1∨G2, H =G1∨G3, E = (G1∨G2)∧G3

and we add the constants 0 y 1.

References

[1] J. Abbott: Implicational algebras. Bull. Math. Soc. Sci. Math. Rpub. Soc.

Roum., Nouv. S´er. 11(59), 3–23 (1968); Errata. Ibid. No.1 (1968).

[2] R. Balbes and P. Dwinger: Distributive Lattices. University of Missouri Press, (1974).

[3] A. Diego: Sur les alg`ebres de Hilbert. Collection de Logique Mathmatique, Sr. A, Fasc. XXI Gauthier-Villars, Paris; E. Nauwelaerts, Louvain (1966).

[4] R. P. Dilworth: Lattices with unique complements. Trans. Am. Math. Soc.

57, (1945) 123–154.

[5] S. Burris and H. Sankappanavar: A course in universal algebra. Graduate Texts in Mathematics,78. Springer-Verlag, New York-Berlin, (1981).

[6] M. Canals Frau and A. V. Figallo: (n+1)-valued modal implicative semilat- tices. Proceedings Of The 22nd International Symposium On Multipe Valued Logic. IEEE Computer Society, 1992, 198–205.

[7] A. V. Figallo: Free Tarski algebras. Preprints del Instituto de Ciencias B´asicas, UNSJ, 2, 1(1997), 11–17.

[8] A. V. Figallo and L. Monteiro: Syst`eme determinant de l’alg`ebre de De Morgan libre sur un ensemble ordonn´e fini. Portugal. Math. 40, 2(1981), no.

2, 129–135.

[9] A. V. Figallo, L. Monteiro and A. Ziliani: Free Three-valued Lukasiewicz, Post and Moisil Algebras over a Poset. IEEE International Symposium on Multiple-Valued Logic 1990. 433–435.

[10] L. Iturrioz and A. Monteiro: C´alculo proposicional implicativo cl´asico con nvariables proposicionales. Rev. de la Uni´on Mat. Argentina, 22(1966), 146.

[11] L. Monteiro: Une construction du r´eticul´e distributif libre sur un ensemble ordonn´e. Colloq. Math.17(1967), 23-27.

[12] L. Monteiro: Une construction des alg`ebres de De Morgan libre sur un en- semble ordonn´e. Rep. Math. Logic, 3(1974), 31–35.

[13] L. Monteiro: Construction des alg`ebres de Tarski libres sur un ensemble ordonn´e. Math. Japon. 23,4(1978), 433–437.

Aldo Figallo, Jr.

Instituto de Ciencias B´asicas,

Universidad Nacional de San Juan, Argentina

&

Departamento de Matem´atica,

Universidad Nacional del Sur, Argentina e-mail: aldofigalllo@gmail.com