BINARY REPRESENTATIONS OF ALGEBRAS WITH AT MOST TWO BINARY OPERATIONS. A CAYLEY THEOREM FOR DISTRIBUTIVE LATTICES

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1 International Journal of Algebra and Computation Vol. 19, No. 1 (2009) c World Scientific Publishing Company BINARY REPRESENTATIONS OF ALGEBRAS WITH AT MOST TWO BINARY OPERATIONS. A CAYLEY THEOREM FOR DISTRIBUTIVE LATTICES YU. M. MOVSISYAN Department of Mathematics, Yerevan State University Yerevan 0025, Armenia yurimovsisyan@yahoo.com Received 24 September 2007 Revised 26 November 2008 Communicated by D. Perrin The notion of binary representation of algebras with at most two binary operations is introduced in this paper, and the binary version of Cayley theorem for distributive lattices is given by hyperidentities. In particular, we get the binary version of Cayley theorem for DeMorgan and Boolean algebras. Keywords: Binary representation; binary Cayley theorem; hyperidentity; semigroup; quasigroup; lattice; distributive lattice; DeMorgan and Boolean algebras. AMS Mathematics Subject Classification: 06D05, 06D30, 06E05, 08A62, 20M30 1. Introduction The binary version of Cayley theorem was first proved for the multiplicative group of a field in [1] (also see [2]). The binary version of Cayley theorem for Boolean algebras is proved in [3] (also see [4 6]). An analogous result is proved for the so-called ternary algebras in [7]. The ordinary Cayley theorem characterizes groups and semigroups in terms of their (faithful) representations by unary transformations (functions). Furthermore, if the initial group or semigroup satisfies an identity, quasi-identity or some formula, then such representation will satisfy (as a unary algebra) the corresponding hyperidentity, conditional hyperidentity or second-order formula. If the initial class of groups or semigroups is not axiomatizable, then such a class is not recognizable by an ordinary Cayley theorem. For example, rings, associative rings, commutative rings, associative-commutative rings, sfields, fields, near-fields or fields of fixed characteristic are axiomatized, but their multiplicative groupoids, multiplicative semigroups and groups are not (Mal tcev, Kogalovski, Sabbach). The situation is analogous for topological rings and topological fields (Pontryagin). 97

2 98 Yu.M.Movsisyan The characterization of these groups and semigroups remains a modern issue in algebra, mathematical logic and topology. On the other hand, having just a unique binary operation (superposition), algebras with two binary operations usually cannot be characterized by unary functions. Having two natural operations (see definitions below) binary transformations (functions) of the type Q Q Q are preferable. It is notable that a considerable role is played by hyperidentities, more specifically by classical binary hyperidentities of idempotency, associativity, distributivity and mediality of the corresponding binary representations. Let us recall that a hyperidentity is a second-order formula of the form: X 1,...,X m x 1,...,x n (w 1 = w 2 ), where w 1, w 2 are words (terms) in the alphabet of functional variables X 1,...,X m and objective variables x 1,...,x n. However hyperidentities are usually presented without universal quantifiers. The hyperidentity w 1 = w 2 is said to be satisfied in the algebra (Q; Σ) if this equality holds whenever every functional variable X i is replaced by an arbitrary operation of the corresponding arity from Σ and every objective variable x j is replaced by an arbitrary element from Q. ThevarietyV of algebras is said to satisfy a certain hyperidentity, if this hyperidentity is satisfied in any algebra of V. In this case the hyperidentity is called a hyperidentity of variety V. For example, if Q( ) is a distributive quasigroup, then the algebra Q(, \,/) satisfies the hyperidentities of distributivity: X(x, Y (y, z)) = Y (X(x, y),x(x, z)), X(Y (x, y),z)=y (X(x, z),x(y, z)). Moreover, if Q(A) is a distributive quasigroup, then the algebra Q(A, A 1, 1 A, 1 (A 1 ), ( 1 A) 1,A ) satisfies these hyperidentities, where A (x, y) =A(y, x), A 1 (x, z) =y A(x, y) =z 1 A(z,y) =x (see [2]). If Q(A) is a medial quasigroup, then the algebra Q(A, A 1, 1 A, 1 (A 1 ), ( 1 A) 1,A ) satisfies the hyperidentity of mediality: X(Y (x, y),y(u, v)) = Y (X(x, u),x(y, v)). 2. Some Preliminary Results on Hyperidentities For the proofs of the following result see [2] (also see [8, 9]). Theorem A. The variety of lattices satisfies the following hyperidentities: X(x, x) =x, (1) X(x, y) =X(y, x), (2)

3 Binary Representations of Algebras 99 X(x, X(y, z)) = X(X(x, y),z), (3) X(Y (X(x, y),z),y(y, z)) = Y (X(x, y),z). (4) Conversely, every hyperidentity of the variety of lattices is a consequence of hyperidentities (1) (4). Theorem B. The variety of modular lattices satisfies the following hyperidentities: (1) (4) and X(Y (x, X(y, z)),y(y, z)) = Y (X(x, Y (y, z)),x(y, z)). (5) Conversely, every hyperidentity of the variety of modular lattices is a consequence of hyperidentities (1) (5). Theorem C. The variety of distributive lattices satisfies the following hyperidentities: (1) (3) and X(x, Y (y, z)) = Y (X(x, y),x(x, z)). (6) Conversely, every hyperidentity of the variety of distributive lattices is a consequence of hyperidentities (1) (3), (6). Theorem D. The variety of Boolean algebras satisfies the following hyperidentities: (1) (3), (6) and F (F (x)) = x, (7) X(F (x),y)=x(f (X(x, y)),y), (8) F (X(F (X(x, y)),f(x(x, F (y))))) = x. (9) Conversely, every hyperidentity of the variety of Boolean algebras is a consequence of hyperidentities (1) (3), (6) (9). Lattices, modular lattices, distributive lattices, Boolean algebras cannot be determined by their hyperidentities [9]. 3. The Concept of Binary Representations of Algebras with One or Two Binary Operations The set of the binary operations defined on the set Q is denoted by FQ 2. If α : A B C and β : A D B are mappings, then define the sum α + β : A D C and the conjugate α : B A C by (α + β)(x, y) =α(x, β(x, y)), α (u, v) =α(v, u), where x A, y D, u B, v A. Ifα : B A C and β : D A B, then define the product α β : D A C by (α β)(x, y) =α(β(x, y),y), where x D, y A.

4 100 Yu.M.Movsisyan Lemma 3.1. (1) The addition α + β is associative. Hence, FQ 2 (+) is a semigroup with a unit element E(x, y) =y. (2) The product α β is associative. Hence, FQ 2 ( ) is a semigroup with unit F (x, y) =x. (3) (α ) = α and (α + β) = α β, (α β) = α + β.hence, the monoids FQ 2 (+) and F Q 2 ( ) are isomorphic. (4) For every fixed set A we obtain two isomorphic categories of sets: Sets A l and Sets A r. The morphisms between sets B and C in SetsA l are mappings of the form A B C, and in Sets A r are mappings of the form B A C. Proof. (1) ((α+β)+γ)(x, y) =(α+β)(x, γ(x, y)) = α(x, β(x, γ(x, y))) = α(x, (β+ γ)(x, y)) = (α +(β + γ))(x, y); (3) (α + β) (x, y) =(α + β)(y, x) =α(y, β(y, x)) = α (β (x, y),y)=(α β )(x, y). Lemma 3.2. The groupoid Q(A) isaquasigroupiffa is invertible in both semigroups F 2 Q ( ) and F 2 Q (+). By denoting E =0andF =1,thenF 2 Q becomes an algebra F 2 Q (+,,, 0, 1) with two binary, one unary, and two nullary operations. Let the set Q, the algebra Γ(+) (not related to Q) with one binary operation be given. The representation of Γ(+) by binary functions of the set Q, or the binary representation of Γ(+) in set Q [1] is a function Γ Q 2 Q which maps every γ Γand(x, y) Q 2 to γ(x, y) Q such that (γ 1 + γ 2 )(x, y) =γ 1 (x, γ 2 (x, y)), for every γ 1,γ 2 Γandx, y Q. If Γ(+) has a unit 0 Γ, then the condition 0(x, y) =y, forx, y Q, is added to the above definition. Suppose such a representation is given. Then to every γ Γ there corresponds a binary operation ϕ(γ) FQ 2 defined by ϕ(γ) :(x, y) γ(x, y). We arrive at the mapping ϕ :Γ FQ 2 with the condition: ϕ(γ 1 + γ 2 )(x, y) =(γ 1 + γ 2 )(x, y) =γ 1 (x, γ 2 (x, y)) = ϕγ 1 (x, ϕγ 2 (x, y)) = (ϕγ 1 + ϕγ 2 )(x, y), i.e. ϕ(γ 1 + γ 2 )=ϕγ 1 + ϕγ 2 and the mapping ϕ is a homomorphism from Γ(+) to FQ 2 (+). Thus, by the representation of Γ(+) by binary functions of set Q, wealso define the homomorphism ϕ :Γ(+) FQ 2 (+). If in addition Γ(+) has a unit 0 Γ, then ϕ(0) = 0, i.e. we obtain a homomorphism between the algebras Γ(+, 0) and FQ 2 (+, 0). The converse is also true: given a homomorphism ϕ :Γ(+) F2 Q (+) or ϕ :Γ(+, 0) FQ 2 (+, 0), then by defining the mapping (γ,x,y) (ϕγ)(x, y) for every γ Γandx, y Q, we obtain a binary representation of Γ in the set Q.

5 Binary Representations of Algebras 101 Now let Γ(+, ) be an algebra with two binary operations. The binary representation of this algebra in Q is defined as a mapping Γ Q 2 Q satisfying: (γ 1 + γ 2 )(x, y) =γ 1 (x, γ 2 (x, y)), (γ 1 γ 2 )(x, y) =γ 1 (γ 2 (x, y),y), for every x, y Q and γ 1,γ 2 Γ. Moreover, if addition or multiplication has a unit element 0 Γor1 Γ, respectively, then the condition 0(x, y) =y or 1(x, y) =x, respectively, for every x, y Q, is added to the definition. We notice, as above, that having a binary representation of Γ(+, ) inthesetq is equivalent to having a homomorphism ϕ :Γ(+, ) FQ 2 (+, ). Moreover, if one or both unit elements exist in Γ(+, ), we have a homomorphism preserving one or both (respectively) unit elements. And finally, let Γ(+,, ) be an algebra with two binary and one unary operations. A binary representation of the algebra Γ(+,, )inthesetqis defined as a mapping Γ Q 2 Q with conditions: (γ 1 + γ 2 )(x, y) =γ 1 (x, γ 2 (x, y)), (γ 1 γ 2 )(x, y) =γ 1 (γ 2 (x, y),y), (γ 1 )(x, y) =γ 1(y, x), for every x, y Q and γ 1,γ 2 Γ. Again, in case + or have a unit element 0 Γ or 1 Γ respectively, the condition 0(x, y) =y or 1(x, y) =x respectively, is added for all x, y Q. Once again, having a binary representation of the algebra Γ(+,, ) in the set Q is equivalent to having a homomorphism ϕ :Γ(+,, ) FQ 2 (+,, ) (and ϕ(0) = 0 or ϕ(1) = 1, provided one or both unit elements exist). The binary representation of an algebra is called faithful, if the corresponding homomorphism is injective. Algebra Γ is said to have a faithful binary representation, if it has a faithful binary representation in a some set Q. Suppose ϕ :Γ FQ 2 is a binary representation of algebra Γ. Then the homomorphic image ϕ(γ) is a subalgebra of the algebra FQ 2. A representation ϕ is said to satisfy an identity (hyperidentity or formula Φ) if that identity (hyperidentity or formula Φ) is satisfied in binary algebra (Q; ϕ(γ)), where ϕ(γ) = {ϕ(γ) γ Γ}. Example. Consider (0, 1) R as a semigroup with the usual product. If Q is a convex subset of a (real or complex) linear space V, then by defining α(x, y) =(1 α)x + αy, α (0, 1), x,y Q, we get a faithful binary representation of the semigroup (0, 1) in Q, satisfying: α(x, y) =(1 α)(y, x), (10) ( ) α(1 β) α(x, β(y, z)) = αβ (x, y),z, (11) 1 αβ

6 102 Yu.M.Movsisyan and the hyperidentities (1), X(Y (x, y),y(u, v)) = Y (X(x, u),x(y, v)), (12) hence satisfying the hyperidentities (6) and X(Y (x, y),z)=y (X(x, z),x(y, z)). (13) In the meantime we get similar definitions of convexors, conrings and conmodules [10 12]. For instance, a convexor on set Q can be defined as a binary representation of the semigroup (0, 1) in Q satisfying the identities (10), (11) and the hyperidentity (1). 4. Binary Representations of Semigroups Proposition 4.1 (Binary Cayley theorem for semigroups). Every semigroup Γ has a faithful binary representation satisfying the hyperidentities: X(x, X(y, z)) = X(X(x, y),x(x, z)), (14) X(X(x, y),x(u, v)) = X(X(x, u),x(y, v)). (15) Proof. Without loss of generality we can suppose that Γ is a monoid. Let Q =Γ and x, y Q, γ Γ. Set γ(x, y) =γ + y, where (+) is the operation of monoid Γ. The condition (γ 1 + γ 2 )(x, y) =γ 1 (x, γ 2 (x, y)) implies from the associativity of addition in Γ. Also, 0(x, y) =0+y = y and the existence of unit 0 Γprovides the faithfulness of the binary representation: γ 1 γ 2 γ 1 (x, 0) γ 2 (x, 0). The aforementioned hyperidentities have a straightforward proof. In case of multiplicative semigroups of associative rings, we obtain an additional hyperidentity (of idempotency). Proposition 4.2 (Binary Cayley theorem for multiplicative semigroups of associative rings). The multiplicative semigroup of associative ring Γ has a faithful binary representation satisfying the hyperidentities (1), (14) and (15). Proof. Without loss of generality we can prove the assertion for associative ring with a unit. Suppose Γ(+, ) is an associative ring with a unit 1 Γ, then by letting Q = Γ and by defining γ(x, y) =(1 γ)x + γy we get the desired faithful binary representation. In case of an associative-commutative ring, the corresponding hyperidentities turn out to be stronger. Corollary 4.1 (Binary Cayley theorem for multiplicative semigroups of associative-commutative rings). The multiplicative semigroup of associativecommutative ring has a faithful binary representation satisfying the hyperidentities (1), (12) (and hence (6), (13)).

7 Binary Representations of Algebras 103 Proposition 4.3 (Binary Cayley theorem for idempotent semigroups). The semigroup Γ is idempotent iff it has a faithful binary representation satisfying the hyperidentities (14), (15), (3) and X(X(y, z),x)=x(x(y, x),x(z,x)), (16) X(X(x, y),x(x, y)) = X(x, y). (17) Proof. Indeed, if Γ(+) has a faithful binary representation satisfying the mentioned hyperidentities, according to (3), (16) and (17) we have (α + α)(x, y) =α(x, α(x, y)) = α(α(x, x),y)=α(α(x, y),α(x, y)) = α(x, y). So Γ is an idempotent semigroup. Proposition 4.4 (Binary Cayley theorem for commutative semigroups). The semigroup Γ is commutative iff it has a faithful binary representation satisfying the hyperidentities (6) and (12). Proof. Indeed, (α + β)(x, y) =α(x, β(x, y)) = β(α(x, x),α(x, y)) = α(β(x, x),β(x, y)) = β(x, α(x, y)) = (β + α)(x, y). Corollary 4.2 (Binary Cayley theorem for semilattices). The semigroup Γ is a semilattice iff it has a faithful binary representation satisfying the hyperidentities (3), (6), (12) and (17) (weak idempotency). 5. Binary Representations of Distributive Lattices, DeMorgan and Boolean Algebras The following result shows that a distributive lattice can be defined by the hyperidentities of its binary representation. Theorem 1 (Binary Cayley theorem for distributive lattices). The algebra Γ(+, ) with two binary operations is a distributive lattice iff it has a faithful binary representation satisfying the hyperidentities (1), (3), (6) and (13). (Compare with Theorem C ). Proof. Without loss of generality we can prove the assertion for bounded distributive lattice. Suppose Γ(+,, 0, 1) is a bounded distributive lattice. Letting Q =Γ and γ(x, y) =γx + y we get a faithful binary representation of Γ(+,, 0, 1) in the set Q satisfying the hyperidentities (1), (3), (6) and (13). Conversely, let us verify the axioms of distributive lattices under the given conditions.

8 104 Yu.M.Movsisyan Idempotency of operations: By setting x = y in the hyperidentity (3) and considering (1), we get α(x, α(x, z)) = α(x, z). (18) Hence, (α+α)(x, y) =α(x, α(x, y)) = α(x, y). Similarly, α(α(x, z),z)=α(x, z) and (α α)(x, y) =α(α(x, y),y)=α(x, y). Commutativity of operations: By setting x = y in the hyperidentity (6) and considering (1), we get and α(x, β(x, z)) = β(α(x, x),α(x, z)) = β(x, α(x, z)) (α + β)(x, y) =α(x, β(x, y)) = β(x, α(x, y)) = (β + α)(x, y). Similarly, the commutativity of product is obtained from the hyperidentity (13). Associativity of operations: This follows from Lemma 3.2. Absorbtion laws: Using (6), (18), (1) and the commutativity of product, we have (α + αβ)(x, y) =α(x, αβ(x, y)) = α(x, α(β(x, y),y)) = α(x, β(α(x, y),y)) = β(α(x, α(x, y)),α(x, y)) = β(α(x, y),α(x, y)) = α(x, y). The second absorbtion law is shows similarly. Distributivity: By definition of sum and product, we have α(β + γ)(x, y) =α((β + γ)(x, y),y)=α(β(x, γ(x, y)),y). Using the commutativity of product, the absorbtion law α + αβ = α, and the hyperidentity (13), we get (αβ + αγ)(x, y) =αβ(x, αγ(x, y)) = βα(x, αγ(x, y)) = β(α(x, αγ(x, y)),αγ(x, y)) = β((α + αγ)(x, y),αγ(x, y)) = β(α(x, y),αγ(x, y)) = β(α(x, y),α(γ(x, y),y)) = α(β(x, γ(x, y)),y). Corollary 5.1 (Binary Cayley theorem for distributive lattices). The algebra Γ(+, ) with two binary operations is a distributive lattice iff it has a faithful binary representation satisfying the hyperidentities (1), (3) and (12). Corollary 5.2 (Binary Cayley theorem for DeMorgan algebras). The algebra Γ(+,,, 0, 1) with two binary, one unary and two nullary operations is a DeMorgan algebra iff it has a faithful binary representation satisfying the hyperidentities (1), (3) and (6).

9 Binary Representations of Algebras 105 Proof. If Γ(+,,, 0, 1) is a DeMorgan algebra, then by letting Q =Γandγ(x, y) = γx + γ y + xy, we obtain the desired faithful binary representation. The converse follows from Theorem 1, since the hyperidentity (13) implies from the hyperidentity (6), for X = α, Y = β. Corollary 5.3 (Dual version). The algebra Γ(+,,, 0, 1) with two binary, one unary and two nullary operations is a DeMorgan algebra iff it has a faithful binary representation satisfying the hyperidentities (1), (3) and (13). The following corollary is equivalent to a result of [3] in the sense that both characterize Boolean algebras by binary functions. Probably, other binary representations of Boolean algebras and distributive lattices are also possible (as in the case of ordinary Cayley theorem), but the formulation of following corollary has the advantage that in it a representation of Boolean algebras rises from the corresponding representation of DeMorgan algebras, and a representation of DeMorgan algebras rises from the corresponding representation of distributive lattices. Corollary 5.4 (Binary Cayley theorem for Boolean algebras). The algebra Γ(+,,, 0, 1) with two binary, one unary and two nullary operations is a Boolean algebra iff it has a faithful binary representation satisfying the hyperidentities (1), (3), (6) and X(x, X(y, x)) = x. Corollary 5.5 (Dual version). The algebra Γ(+,,, 0, 1) with two binary, one unary and two nullary operations is a Boolean algebra iff it has a faithful binary representation satisfying the hyperidentities (1), (3), (13) and X(X(y, x),y)=y. Similar results hold for distributive q-lattices introduced in [13]. Open problems: (1) Characterize (modular) q-lattices by binary representations. (2) Obtain necessary and sufficient conditions for characterization of multiplicative semigroups of associative (associative-commutative) rings by binary representations. (3) Characterize the hyperidentities of variety of (associative) rings. For example, the following hyperidentity is satisfied in every ring: X(X(Y (x, x),y(x, x)),y(x(x, x),x(x, x))) = X(Y (X(x, x),x(x, x)),x(y (x, x),y(x, x))). Historical notes: Addition and multiplication on the set FQ 2 was considered in [14, 15] in relation with orthogonality of Latin squares. Hyperidentities of associativity and distributivity was first considered in binary algebras with quasigroup operations in [15,16]. The general concept of hyperidentity in algebras is studied in monographs [17, 18]. Let J (A) be the term (or polynomial) algebra of the algebra A. Wecallthe hyperidentities of algebra J (A) term hyperidentities of A. Term hyperidentities were first considered in [19, 20], and studied in monographs [21, 22].

10 106 Yu.M.Movsisyan Many related papers can be found in these references. References [1] Yu. M. Movsisyan, The multiplicative group of a field and hyperidentities, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989) [English transl. in Math. USSR Izvestiya 35 (1990) ]. [2] Yu. M. Movsisyan, Hyperidentities in algebras and varieties, Uspekhi Mat. Nauk 53(1) (1998) [English transl. in Russian Math. Surv. 53(1) (1998) ]. [3] S. L. Bloom, Z. Esik and E. G. Manes, A Cayley theorem for Boolean algebras, Amer. Math. Manthly 97 (1990) [4] G. Bergman, Actions of boolean rings on sets, University of California, Berkely, CA, 94720, preprint. [5] I. Chajda, A representation of the algebra of quasiordered logic by binary functions, Demonstratio Math. XXVII (1994) [6] J. A. Brzozowski, A characterization of de Morgan algebras, Int. J. Algebra Comput. 11(5) (2001) [7] Z. Esik, A Cayley theorem for ternary algebras, Int. J. Algebra Comput. 8(3) (1998) [8] Yu. M. Movsisyan, Hyperidentites of Boolean algebras, Izv. Ross. Akad. Nauk, Ser. Mat. 56 (1992) [English transl. in Russian Acad. Sci. Izv. Math. 40(3) (1993) ]. [9] Yu. M. Movsisyan, Algebras with hyperidentites of the variety of Boolean algebras, Izv. Ross. Akad. Nauk, Ser. Mat. 60 (1996) [English transl. in Russian Acad. Sci. Izv. Math. 60 (1996) ]. [10] L. A. Skornjakov, Convexors, Studia Sci. Math. Hungar. 16 (1981) [11] L. A. Skornjakov, Stochastic acts and conmodules, Semigroup Forum 25 (1982) [12] A. Romanowska and J. D. H. Smith, Modes (World Scientific, 2002). [13] I. Chajda and M. Kotrle, Subdirectly irreducible and congruence q-lattices, Czech. Math. J. 43 (1993) [14] S. K. Stein, On the foundation of quasigroups, Trans. Amer. Math. Soc. 85 (1957) [15] V. D. Belousov, Systems of quasigroups with generalized identities, Uspekhi Mat. Nauk. 20 (1965) (1967); [English transl. in Russian Mathematical Surveys 20 (1965) ]. [16] J. Aczel, Proof of a theorem on distributive type hyperidentities, Algebra Univers. 3 (1971) 1 6. [17] Yu. M. Movsisyan, Introduction to the Theory of Algebras with Hyperidentities (Yerevan State University Press, Yerevan, 1986) (in Russian). [18] Yu. M. Movsisyan, Hyperidentities and Hypervarieties in Algebras (Yerevan State University Press, Yerevan, 1990) (in Russian). [19] W. Taylor, Hyperidentities and hypervarieties, Aequationes Math. 23 (1981) [20] G. Bergman, Hyperidentities of groups and semigroups, Aequationes Math. 23 (1981) [21] K. Denecke and Sh. L. Wismath, Hyperidentities and Clones (Gordon and Breach Science Publishers, 2000). [22] J. Koppitz and K. Denecke, M-Solid Varieties of Algebras (Springer, 2006).