Notes on the Variety of Ternary Algebras

In this work we review the class  of ternary algebras introduced by J. A. Brzozowski and C. J. Serger in [1]. We determine properties of the congruence lattice of a ternary algebra A. The most important result refers to the construction of the free ternary algebra on a poset. In particular, we describe the poset of the join irreducible elements of the free ternary algebra with two free generators.


Introduction and Preliminaries
Different algebras began to be used in the thirties as a tool for the analysis and design of digital circuits.One of the pioneers in this field was C. E. Shannon, who in 1938 applied Boolean algebras in his work " A Symbolic Analysis of Change and Retransmission circuits".Even though technology has changed substantially, these algebras still play a very important role in this field.In contrast, the use of ternary algebras has been much more limited, and the first applications of trivaluada logic to switching circuits were made by Goto in 1948.He worked with the algebra { } 0,1 2,1 , , , , 0,1 , of type ( ) 2, 2,1, 0, 0 , where , 1 a a = − .He interpreted 1 2 as an indefinite value, and used it to represent a transition state between 0 and 1.In addition, by means of the resolution of trivalued logic equations, he managed to derive the necessary conditions to prevent hazardous behavior in sequential relay contact networks.In 1959, Muller worked with this same ternary algebra, and applied it to the study of the transient phenomena in switching circuits, and observed, among other things, that it satisfies Kleene's law: ( ) ( ) ( ) In 1964, Yoeli and Rinon applied them to the study of static hazards in combinational switching circuits.In 1965, Eichelberger used them in the analysis of hazard phenomena in combinational and sequential switching circuits.
In 1983, Mukaidono studied a special class of ternary functions, the regular ones, and indicated a set of axioms to define a ternary algebra, calling the resulting algebra Kleene algebra with a centre.However, it was not until 1995 that Brozozowski and Serger presented them as a De Morgan algebra with an additional constant, e , which satisfies these identities: 1) e e = , 2) ( ) a a e a a + + = + .The same year Negulescu introduced the process spaces, a formalism which allows the modelling of the behaviour of interacting systems.
In 1996, Brozozowski, Lou and Negulescu considered certain subalgebras of the process spaces, i.e. the subset-pair algebras, and showed that each finite ternary algebra is isomorphic to a subset-pair algebra.In 1997, Esik showed that this property is valid for any ternary algebra and that, in general, each ternary algebra is isomorphic to an algebra of ternary functions, this result being analogous to Caley's representation for Boolean algebras.
This class of algebras is applied in the detection of hazards in the combinational circuits, in the modelling of interacting systems by means of process spaces and in the analysis of asynchronous sequential circuits by means of Eichelberger's method of ternary simulation, among other applications.
This article is organized as follows.Firstly, we summarize the main definitions and necessary results for a better comprehension of the article.In Section 2, we characterize the subdirectly irreducible algebras, examine the congruence lattice of a ternary algebra and establish properties of the variety  .Finally, in Section 4, we indicate a construction of the free ternary algebra on a poset, and apply it to the free ternary algebra on one free generator and describe the poset of the join irreducible elements of the free ternary algebra on two free generators.
(T5) ( ) ( ) a a b b a a + ∼ + ⋅ ∼ = + ∼ for all , a b L ∈ .(T6) In each ternary algebra the only element that satisfies the condition x x ∼ = is e .(T7) Every finite ternary algebra has an odd number of elements.(T8) There exist, up to isomorphism, one ternary algebra with five elements, two with seven, and four with nine.
(T9) For each odd natural number 3 n ≥ , there exists at least one ternary algebra with n elements.Clearly, (T4) is the dual property of (T2) while (T5) is equivalent to (T2).Two well known examples of ternary algebras, of significant importance for the rest of this work, are:

The Congruence Lattice and Subdirectly Irreducible Algebras
In order to characterize the subdirectly irreducible ternary algebras we first observe that the ternary algebras arises from Kleene algebras endowed with a nullary operation.Definition 2.1 A Kleene algebra , , , , 0,1 L + ⋅ ∼ is a De Morgan algebra which satisfies the property ( ) We represent this class of algebras by C  .
Remark 2.1 From Definitions 2.1 y 2.2 we have that: 1) If , , , , 0, ,1 L e + ⋅ ∼ ∈  , then the reduct ˆ, , , , 0,1 The preceding results and some from universal algebra allow us to ensure that: Theorem 2.2 1) The simple algebras in this variety coincide with the subdirectly irreducible.
2) Variety  is semisimple, residually finite and locally finite.Besides, with respect to the directly indecomposable algebras we can state that: Theorem 2. 3 In  there exist directly indecomposable algebras which are not subdirectly irreducible.Proof.Any ternary algebra p T with p a prime number greater than or equal to 3 is directly indecomposable but not subdirectly irreducible.□ From the above results we can conclude that: Corollary 2.2 Variety  is not directly representable.

Lattice ( )
Con  L Now we will consider the congruence lattice of a ternary algebra L , ( ) , and will establish some of its most relevant properties.
Proof.In the ternary algebra 5 T , the congruences ( ) The equations that define the principal congruences in the De Morgan algebras obtained by H. P. Sankappanavar in [2] allows the following: Theorem 2.7 Variety  has equationally definable principal congruences.
From the results obtained by A. Day in 1971 [3], and for W. J. Blok y D. Pigozzi in 1982 [4], we can conclude that: Theorem 2.8 1) Variety  is congruence-distributive, it has the congruence extension property and the principal congruence extension property. 2) A B∈  , being B a subalgebra of A and , a b B ∈ .In addition, taking into account the characterization given by P. Kholer y D. Pigozzi in 1980 [5], we have that: Theorem 2.9 For all L ∈  , ( ), ,

Free Ternary Algebras over an Poset
In 1945, R. Dilworth [6] introduced the notion of free lattice over a poset.Later on, this subject was studied by many authors and in different classes of algebras.In this section we indicate a construction of the free ternary algebra over a poset, a result that generalizes those obtained by R. Balbes over free ternary algebras [7].
Let  be a variety of similar algebras which have an underlying ordered structure definable by a set of equ- ations ( ) ( ) , , i i p x y q x y = , 1 i n ≤ ≤ , which are in terms of the algebra operations.The notion of free algebra over an poset I on  can be defined as follows: Definition 3.1 A  -algebra ( )

L I
 is free over I if it satisfies the conditions that follow: (L1) there exists an order-embedding ( )  generated by X , (L2) for each  -algebra A and each increasing function : f I A → , there exists a homomorphism ( ) . Remark 3.1 It is clear that: 1) If the free algebra over an poset I exists, it is unique up to isomorphism.
2) The homomorphism h is unique.
3) If I is an poset of cardinality n , 0 n > , and the order over I is the identity, then ( ) is the free algebra with a set of n free generators.
4) In variety  each algebra has a underlying ordered structure defined by the equation x y  if, and only if, x x y = ⋅ .In what follows, we determine the structure of ( )

Construction of the Free Ternary Algebra over a Poset I
A. Figallo Jr. and A. Ziliani in [8] construct the free algebra over a poset in finitely generated varieties; by using a similar argument, we indicate a construction of the free ternary algebra over a poset.
We consider a poset I , the set E of the increasing functions of I in the ternary algebra 0 T and the function 0 : for each f E ∈ and for each i I ∈ .Theorem 3.1 The free ternary algebra ( ) From what has been stated, we conclude that i j  on I if, and only if, i j G G  on G , from which it turns out that ( ) . Therefore, (L1) holds.
(L2) Let L be a ternary algebra and : f I L → an increasing function, by Corollary 2.1, L is isomorphic to a subalgebra of 0 X T being X an arbitrary set, then there exists a subalgebra S of 0 X T and an isomorphism : L S ϕ → .We claim that there is a homomorphism

Free Ternary Algebra on One Free Generator
If L is a ternary algebra and S L In [7], R. Balbes proves that [ ] 1 : and 0 , we apply this result and those obtained before in order to get the free ternary algebra on one free generator.

{ }
0 = where 1 2 3 , , f f f are defined as follows ( ) Some of the elements of 0 E T are specified in the Table 1.Then, the free ternary algebra on one free generator is: , , , , , , , , , , 3 3  which coincides with the result obtained by R. Balbes en [7].
The algebra

The Poset of the Join Irreducible Elements of the Free Ternary Algebra on Two Free Generators
In [7] R. Balbes proves that the poset of the join irreducible elements of the free ternary with a finite of free generators X is isomorphic with ( ) ( ) ordered by the relation T S  if, and only if, S T ⊆ .We apply this result and those obtained above in order to determine the poset of the join irreducible elements of a free ternary algebra on two free generators.We consider the set : : is increasing , , , , , , , , where each of the functions are defined in the  ( ) for each i I ∈ .In the Table 3, we specify some of the elements of 0 E T .
The poset IP coincides with that obtained by R. Balbes in [7] and it is the one indicated in Figure 2.

Proof.
It is sufficient to consider the equivalence classes [ ] considered in the proof of the theorem 2.4.□ Theorem 2.6 Variety  is not congruence-regular.Proof.In the ternary algebra 7 T , it is sufficient to consider the congruences As a direct consequence of Theorems 2.4 y 2.5, we can formulate: Corollary 2.3 Variety  is neither arithmetic nor discriminatory.
the set of the finitely generated congruences of the algebra L .

⊆
, we denote by [ ] S  the ternary algebra generated by the set S and by [ ] S  the free ternary algebra with S as the set of free generators, and we set

Figure 1 .
Figure 1.The free ternary algebra on one free generator.

Figure 2 .
Figure 2. The poset of join irreducible elements of the free ternary algebra on two free generators.
The only subdirectly irreducible centered Kleene algebra is the three-elements chain.As a direct consequence of Theorem 2.1, we can establish Corollary 2.1 The only subdirectly irreducible ternary algebra is

Table 2 .
The elements of E.