The connection of skew Boolean algebras and discriminator varieties to Church algebras

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1 The connection of skew Boolean algebras and discriminator varieties to Church algebras Karin Cvetko-Vah and Antonino Salibra Abstract. We establish a connection between skew Boolean algebras and Church algebras. We prove that the set of all semicentral elements in a right Church algebra forms a right-handed skew Boolean algebra for the properly defined operations. The main result of this paper states that the variety of all semicentral right Church algebras of type τ is term equivalent to the variety of right-handed skew Boolean algebras with additional regular operations. As a corollary to this result we show that a pointed variety is a discriminator variety if and only if it is a 0-regular variety of right-handed skew Boolean algebras. 1. Introduction Skew Boolean algebras are non-commutative one-pointed generalisations of (possibly generalised) Boolean algebras. A typical example of a skew Boolean algebra is provided by the algebra of all partial functions from a given set A to {0, 1}, with properly defined operations. Just as Boolean algebras have a lattice reduct, skew Boolean algebras have a skew lattice reduct, to be defined below. By a result of Leech [11] every skew lattice S embeds into the direct product of a right-handed and a left-handed skew lattice, called the right and left factors of S (See Section 2.3 for definitions of right- and left-handedness). It follows that right- and left-handed skew lattices are not just special cases of skew Boolean algebras but rather reveal significant structures. For instance, it was proven in [5] that a skew lattice S satisfies an equational identity if and only if both the left and the right factor of S satisfy that equational identity. The significance of skew Boolean algebras is revealed in part by a result of Leech [12] stating that any right-handed skew Boolean algebra can be embedded into some skew Boolean algebra of partial functions. This result has been further explored in [1] and [9], where results show that any skew Boolean algebra is dual to a sheaf over a locally-compact Boolean space. Skew Boolean algebras are closely related to pointed discriminator varieties, i.e. discriminator varieties with a constant term. To see this, one needs skew Boolean intersection algebras, i.e. skew Boolean algebras that are closed under finite infima with respect to the natural partial order which is defined precisely in the next section. We denote by P D 0 the pointed discriminator variety of type (3, 0) generated by the class of all pointed discriminator algebras 2010 Mathematics Subject Classification: Primary: 03G10; Secondary: 08B26. Key words and phrases: skew Boolean algebra, Church algebra, discriminator variety, factor congruence, decomposition operator.

2 2 K. Cvetko-Vah and A. Salibra Algebra univers. (A; d, 0), where d is the discriminator function on A and 0 is a nullary operation. By a result of Bignall and Leech [3] the variety P D 0 is term equivalent to the variety of all right-handed skew Boolean intersection algebras. The key observation motivating the introduction of Church algebras [14] is that many algebras arising in completely different fields of mathematics - including Heyting algebras, rings with unit, or combinatory algebras - have a term operation q satisfying the fundamental properties of the if-then-else connective: q(1, x, y) = x and q(0, x, y) = y. As simple as they may appear, these properties are enough to yield rather strong results. We mention here that central elements in the sense of Vaggione [18] are especially convenient to work with because they can be given a simple equational characterisation in Church algebras. A central element a of a Church algebra satisfies the identities q(a, a, 0) = a = q(a, 1, a) and determines a pair (θ a, θ a ) of complementary factor congruences. The identity q(a, a, 0) = a implies that θ a equals the principal congruence θ(a, 0) and, similarly, the identity q(a, 1, a) = a implies θ a = θ(a, 1). We also mention that central elements of a Church algebra constitute a Boolean algebra with respect to the operations x y = q(x, y, 0), x y = q(x, 1, y) and x = q(x, 0, 1). In [16] the notion of a Boolean-like algebra was introduced as a generalisation of a Boolean algebra. A Boolean-like algebra is a Church algebra A such that every a A is central. Boolean-like varieties can be characterised in terms of discriminator varieties: a double-pointed variety V of a given type τ is a Boolean-like variety iff V is a discriminator variety such that A = 2 for every subdirectly irreducible member A of V. It also turns out that some important properties of Boolean algebras are shared not only by Boolean-like algebras, but also by algebras whose elements satisfy all the equational conditions of central elements except x = q(x, 1, x). These algebras, and the varieties they form, were termed idempotent semi- Boolean-like in [16]. Idempotent semi-boolean-like algebras are closely related to double-pointed discriminator varieties. Actually, a double-pointed variety is a discriminator variety iff it is idempotent semi-boolean-like and 0-regular [16, Theorem 5.6]. In the present paper we study how skew Boolean algebras are connected to Church algebras. Our aim is to compare the characterisation of one-pointed discriminator varieties by Bignall and Leech [3] to the characterization of double-pointed discriminator varieties by Salibra et al. [16]. In Section 3 we introduce the one-pointed version of Church algebras. We define a right (resp. left) Church algebra as an arbitrary algebra of a given type τ with a fixed ternary term q and a constant 0 that satisfy the identity q(0, x, y) = y (resp. q(0, x, y) = x). We generalise the notation of a central element and introduce the notion of a semicentral element that satisfies all the equational conditions of central elements except q(x, 1, x) = x. In Section 4 we introduce semicentral right Church algebras as right Church algebras with all elements being semicentral. We show that operations can be defined on the set S(A)

3 Vol. 00, XX The connection of skew Boolean algebras to Church algebras 3 of all semicentral elements in a right Church algebra A such that S(A) with these operations forms a right-handed skew Boolean algebra. We prove that the variety of all semicentral right Church algebras of type τ is term equivalent to the variety of right-handed skew Boolean algebras with additional regular operations. In Section 5 we study the connection with discriminator varieties and prove that a pointed variety is a discriminator variety if and only if it is a 0-regular variety of semicentral right Church algebras. New systems of axioms for skew Boolean intersection algebras are provided. 2. Preliminaries If A is an algebra and x, y A, then θ(x, y) denotes the least congruence on A including the pair (x, y). We denote respectively by, the least and the greatest congruence of the congruence lattice Con(A) Factor congruences. Definition 1. A congruence φ on an algebra A is a factor congruence if there exists another congruence φ such that φ φ = and φ φ =. In this case we call (φ, φ) a pair of complementary factor congruences. Under the hypotheses of Definition 1 the homomorphism f : A A/φ A/ φ defined by f(x) = (x/φ, x/ φ) is an isomorphism. Hence, we have: (φ, φ) is a pair of complementary factor congruences of A if, and only if, A = A/φ A/ φ under the natural map x (x/φ, x/ φ). So, the existence of factor congruences is just another way of saying this algebra is a direct product of simpler algebras. The set of factor congruences of A is not, in general, a sublattice of Con(A). and are the trivial factor congruences, corresponding to A = A B, where B is a trivial algebra; of course, B is isomorphic to A/ and A is isomorphic to A/. An algebra A is directly indecomposable if it admits only the two trivial factor congruences ( and ). Clearly, every simple algebra is directly indecomposable, while there are algebras which are directly indecomposable but not simple: they have congruences, which however do not split the algebra up neatly as a Cartesian product Decomposition operations. Factor congruences can be characterised in terms of certain algebra homomorphisms called decomposition operations (see [15, Def. 4.32] for more details). Definition 2. Let A be an algebra of type τ. A decomposition operation on A is a function f : A A A satisfying the following conditions: f(x, x) = x; f(f(x, y), z) = f(x, z) = f(x, f(y, z)); f is an algebra homomorphism from A A into A.

4 4 K. Cvetko-Vah and A. Salibra Algebra univers. When τ =, the third condition of Definition 2 is redundant. There exists a bijective correspondence between pairs of complementary factor congruences and decomposition operations, and thus, between decomposition operations and factorisations like A = B C. Proposition 1. [15, Thm. 4.33] Let A be an algebra of type τ. Given a decomposition operation f on A, the binary relations θ f and θ f defined by: x θ f y if, and only if, f(x, y) = x, x θ f y if, and only if, f(x, y) = y, form a pair of complementary factor congruences. Conversely, given a pair (φ, φ) of complementary factor congruences, the function f defined by: f(x, y) = u if, and only if, yφu φx, (2.1) determines a decomposition operation on A such that φ = θ f and φ = θ f. Notice that if (φ, φ) is a pair of complementary factor congruences, then for all x and y there is just one element u such that yφu φx Skew Lattices. We review some basic definitions and results on a noncommutative generalisation of lattices, probably the most interesting and successful: the concept of skew lattice [11], in fact, along with the related notion of skew Boolean algebra, has important connections with discriminator varieties; the interested reader is referred to [13] or [17] for far more comprehensive accounts. Definition 3. A skew lattice is an algebra A = (A,, ) of type (2, 2) satisfying: Associativity: x (y z) = (x y) z; Idempotence: x x = x = x x Absorption: x (x y) = x = x (x y); x (y z) = (x y) z (y x) x = x = (y x) x It is not difficult to see that the absorption condition is equivalent to the following pair of biconditionals: x y = y iff x y = x; and x y = x iff x y = y. Definition 4. A skew lattice A is called left-zero ( right-zero) if a b = a (a b = b) for all a, b A. Skew lattices that are as noncommutative as possible are rectangular skew lattices. They include left-zero (right-zero) skew lattices and other remarkable examples. Definition 5. A skew lattice is rectangular if it satisfies the identity x y = y x. We collect in the next lemma and propositions some useful properties of rectangular skew lattices.

5 Vol. 00, XX The connection of skew Boolean algebras to Church algebras 5 Lemma 1. The following conditions are equivalent for a skew lattice A: (1) A is rectangular. (2) A satisfies x y x = x. (3) A satisfies x (y x) = y x. (4) A satisfies (y x) y = y x. (5) A satisfies x y x = x. (6) A satisfies (x y) x = x y. (7) A satisfies y (x y) = x y. (8) A satisfies x y z = x z. (9) A satisfies x y z = x z. Proposition 2. Every rectangular skew lattice A is isomorphic to a direct product of a left-zero skew lattice and a right-zero skew lattice. Proof. Define X = {x A : x A} and Y = {A x : x A}. It is not difficult to show that A can be embedded into X Y via the mapping f(x) = (x A, A x). Every rectangular skew lattice is completely distributive: the identities x (y z) = (x y) (x z) and (y z) x = (y x) (z x) are satisfied for {, } = {, }. Lemma 2. In any skew lattice the relation, defined by a b iff a b = a = b a or equivalently a b = b = b a, is a partial ordering. Observe that a b a a a b a for every a, b. Define x = {y : y x} and x = {y : y x}. It can be seen that x = {x y x : y A} and x = {x y x : y A} are subalgebras of A, by means of which we can identify further interesting subclasses of skew lattices: Definition 6. A skew lattice A is normal if x is a lattice for every x; it is Boolean if x is a Boolean lattice for every x A. Normal skew lattices are a variety, axiomatized relative to skew lattices by the equation x y z x = x z y x. In any skew lattice A we define a preorder: a d b iff a b a = a iff b a b = b. Observe that a b d a d a b and b a d a d b a, for every a, b. The equivalence induced by d, denoted as D, is in fact a congruence, and A/D is the maximal lattice image of A. The D-equivalence class D a of an element a is {a} iff, for all b A, a b = b a. We remark that D a D b in A/D iff a b a = a in A.

6 6 K. Cvetko-Vah and A. Salibra Algebra univers. Proposition 3. Let A be a skew lattice. (1) If a, b D x then a b, a b D x. (2) If B is a rectangular subalgebra of A, then adb for all a, b B. (3) For every a, D a is a maximal rectangular subalgebra of A. Proposition 4 ([11]). Let A be a skew lattice and let x, y be members of A/D. (1) If x y, then for each a x there exists b y such that a b in A, and dually, for each b y there exists a x such that a b in A. (2) Let c x A/D y be given and pick a x and b y such that c a, b in A. Then c = a b = b a in A. Thus: x A/D y = {a b : a x, b y, and a b = b a}. Similar considerations hold for A/D. We have that: Proposition 5. Every skew lattice is regular, i.e., it satisfies the identities x y x z x = x y z x and x y x z x = x y z x. Two further preorders can be defined on a skew lattice: (1) x l y iff x y = x; (2) x r y iff y x = x. The equivalences L and R, respectively induced by l and r, are again congruences. Definition 7. A skew lattice is left-handed ( right-handed) if L = D (R = D). L (R) is the minimal congruence making A/L (A/R) a right-handed (lefthanded) skew lattice. Lemma 3. The following conditions are equivalent for a skew lattice A: (1) A is right-handed; (2) for all a, b A, adb implies a b = b; (3) for all a, b A, a b a = b a Skew Boolean algebras. If we expand the skew lattice signature by adjoining a subtraction operation and a constant 0, we get the following noncommutative variant of Boolean algebras (see [12]). Definition 8. A skew Boolean algebra ( SBA, for short) is an algebra A = (A,,, \, 0) of type (2, 2, 2, 0) such that: its reduct (A,, ) is a normal skew lattice satisfying the identities x (y z) = (x y) (x z) and (y z) x = (y x) (z x). Both x 0 = 0 = 0 x and x 0 = x = 0 x hold. the operation \ satisfies the identities (x y x) (x \ y) = x = (x \ y) (x y x); x y x (x \ y) = 0 = (x \ y) x y x.

7 Vol. 00, XX The connection of skew Boolean algebras to Church algebras 7 The commutative SBAs are precisely the class of all generalised Boolean algebras. Recall that SBAs satisfy the generalised De Morgan laws: x \ (y z) = (x \ y) (x \ z) and x \ (y z) = (x \ y) (x \ z). An SBA is primitive if it consists of exactly two D-classes: A > {0}, where A is a rectangular skew lattice. Of interest here are right-handed cases, whose operations have the following descriptions for x, y 0: 0 y 0 0 y x x x 0 y x 0 y \ 0 y x x 0 Particularly important are the right-handed SBAs 3 R and its subalgebra 2 R with respective D-class structures {1, 2} > {0} and {1} > {0}. To within isomorphism, 3 R and 2 R are the only nontrivial subdirectly irreducible righthanded SBAs. Thus every right-handed SBA is a subdirect product of copies of 3 R and/or 2 R. As a consequence an equation is an identity on all right-handed SBAs if and only if it holds on 3 R. In general, the right-handed primitive SBAs are the directly indecomposable right-handed SBAs. Lemma 4 below yields some further easily verified identities that simplify computation in right-handed SBAs. Lemma 4. Let A be a right-handed SBA and x, y, z A. Then: (1) (x y) \ z = (x \ z) (y \ z), (2) (x y) \ z = (x \ z) (y \ z), (3) x (y \ z) = (x y) \ (x z), (4) (x \ y) \ z = (x \ z) \ (y \ z), (5) x \ (y \ z) = (x \ y) (x z x). Proof. All five equations are easily seen to hold on 3 R. Thus they hold on right-handed SBAs. Lemma 5. For right-handed SBAs the following are equivalent: (1) xdy; (2) x y = y & y x = x; (3) x y = x & y x = y; (4) x \ y = 0 & y \ x = 0; (5) z, x z = y z; (6) z, z \ x = z \ y. SBAs such that every finite subset of their universe has an infimum w.r.t. the underlying natural partial ordering of the algebra stand out for their significance. We denote the infimum of x and y w.r.t. the natural partial order by x y and refer to the operation as intersection in order to distinguish it from the skew lattice meet. It turns out that SBAs with the additional

8 8 K. Cvetko-Vah and A. Salibra Algebra univers. operation can be given an equational characterisation provided we include the operation into the signature. Definition 9. A skew Boolean -algebra is an algebra (A;,,, \, 0) of type (2, 2, 2, 2, 0) s.t.: the reduct (A;,, \, 0) is an SBA and the reduct (A; ) is a meet semilattice; A satisfies the identities x (x y x) = x y x; x (x y) = x y = (x y) x. The next theorem by Bignall and Leech [3] provides a powerful bridge between the theories of SBAs and pointed discriminator varieties: Theorem 1. (i) The variety of type (3, 0) generated by the class of all pointed discriminator algebras (A; t, 0), where t is the discriminator function on A and 0 is a constant, is term equivalent to the variety of right handed skew Boolean -algebras. (ii) Every discriminator variety is term equivalent to a variety V of right handed skew Boolean -algebras with additional operations (g i ) i I such that, for all algebras A V, every congruence θ on the term reduct (A; t, 0) is compatible with g i for all i I. 3. Right (Left) Church algebras We recall that a Church algebra [14] is a double-pointed algebra with a term operation q satisfying the fundamental properties of the if-then-else connective: q(1, x, y) = x and q(0, x, y) = y. To compare Church algebras and SBAs, in the following definition we introduce the one-pointed counterpart of Church algebras. Definition 10. A right (resp. left) Church algebra (RCA) (resp. LCA) is an algebra A of type τ, where we fix a ternary term q and a constant 0 such that q(0, x, y) = y (resp. q(0, x, y) = x). Of course, every pointed algebra is an RCA (LCA), because the third (second) projection map is a term. The notion of an RCA (LCA) is a terminological convenience, to be supplemented by further demands in the main results of the paper. A 0 = (A, q, 0) is the pure reduct of A. The following right (resp. left) derived operations will be important in the remaining part of the paper: x y = q(x, x, y) (resp. x y = q(y, x, y)); x y = q(x, y, 0) (resp. x y = q(y, 0, x)); y \ x = q(x, 0, y) (resp. y \ x = q(x, y, 0)).

9 Vol. 00, XX The connection of skew Boolean algebras to Church algebras 9 Without loss of generality, in the following we develop the theory of RCAs. An element x of an RCA is idempotent if x x = x. Example 1. The following are examples of RCAs: 1. Lattices with bottom, where q(x, y, z) = (x y) z. 2. SBAs, where q(x, y, z) = (x y) (z \ x) (see Section 4.1). 3. Rings, where q(x, y, z) = xy + z xz (see Section 6) Factor Elements. An element e of an RCA is a factor element if the binary operation q(e,, ) is a decomposition operation. If e is a factor element, then by Proposition 1 the relations θ e = {(x, y) : q(e, x, y) = x} and θ e = {(x, y) : q(e, x, y) = y} constitute a pair of complementary factor congruences of A. We denote by F(A) the set of factor elements of an RCA A. The following trivial proposition equationally characterises the factor elements. Proposition 6. A factor element e satisfies the following identities for all x, y, z, x, y A: (A1) q(e, x, x) = x. (A2) q(e, q(e, x, y), z) = q(e, x, z) = q(e, x, q(e, y, z)). (A3) q(e, g(x 1,..., x n ), g(y 1,..., y n )) = g(q(e, x 1, y 1 ),..., q(e, x n, y n )), for every g τ. Of interest is the fact that the set of all factor elements of an RCA forms a subalgebra of the pure reduct. Proposition 7. Let A be an RCA. Then F(A) is a subalgebra of the pure reduct A 0. Proof. Let a, b, c be factor elements and let d = q(a, b, c). We show that d is also a factor element. q(d, x, x) = q(d, q(a, x, x), q(a, x, x)) by (A1) = q(a, q(b, x, x), q(c, x, x)) by (A3) = q(a, x, x) by (A1) = x by (A1) q(d, q(d, x, y), z) = q(d, q(a, q(b, x, y), q(c, x, y)), z) by (A3) = q(a, q(b, q(b, x, y), z), q(c, q(c, x, y), z)) by (A3) = q(a, q(b, x, z), q(c, x, z)) by (A2) = q(d, x, z) by (A3) Similarly for q(d, x, q(d, y, z)) = q(d, x, z).

10 10 K. Cvetko-Vah and A. Salibra Algebra univers. In the following we apply in each line axiom (A3). q(d, g( x), g(ȳ)) = q(a, q(b, g( x), g(ȳ)), q(c, g( x), g(ȳ))) = q(a, g(..., q(b, x i, y i ),... ), g(... q(c, x i, y i ),... )) = g(..., q(a, q(b, x i, y i ), q(c, x i, y i )),... ) = g(..., q(d, x i, y i ),... ) Proposition 8. Let A be an RCA, e A be a factor element, and s e, t e : A A be two maps defined by t e (x) = e x and s e (x) = x \ e. Then the following conditions hold: (1) t e and s e are endomorphisms of the pure reduct A 0. (2) ker(s e ) = θ e and ker(t e ) = θ e. Proof. (1) The map t e is a homomorphism: t e (q(x)) = e q(x) = q(e, q(x), 0) by def of = q(e, q(x), q(0)) by q(0, 0, 0) = 0 = q(q(e, x 1, 0), q(e, x 2, 0), q(e, x 3, 0)) by (A3) = q(t e (x 1 ), t e (x 2 ), t e (x 3 )). The proof is similar for the map s e. (2) We show that ker(t e ) = {(x, y) : q(e, x, y) = y}. Assume t e (x) = t e (y). Then q(e, x, y) = q(e, q(e, x, 0), y) by (A2) = q(e, q(e, y, 0), y) by e x = e y = q(e, y, y) by (A2) = y by (A1) In the opposite direction, let q(e, x, y) = y. Then we have: q(e, y, 0) = Hp. q(e, q(e, x, y), 0) = (A2) q(e, x, 0). We show that ker(s e ) = {(x, y) : q(e, x, y) = x}. Assume s e (x) = s e (y). Then q(e, x, y) = q(e, x, q(e, 0, y)) by (A2) = q(e, x, q(e, 0, x)) by x \ e = y \ e = q(e, x, x) by (A2) = x by (A1) In the opposite direction, let q(e, x, y) = x. Then we have: q(e, 0, x) = Hp. q(e, 0, q(e, x, y)) = (A2) q(e, 0, y).

11 Vol. 00, XX The connection of skew Boolean algebras to Church algebras Semicentral Elements. We recall from [16] that an element e of a Church algebra is central if it is a factor element (in the terminology of Section 3.1) satisfying the identity q(e, 1, 0) = e. This last condition can be split into two identities, the first one expressed in terms of the constant 0 and the second one in terms of the constant 1. Proposition 9. Let e be a factor element of a Church algebra A = (A, q, 0, 1). Then the following conditions are equivalent: (1) q(e, 1, 0) = e; (2) q(e, e, 0) = e and q(e, 1, e) = e; Proof. (1) (2). q(e, e, 0) = q(e, q(e, 1, 0), 0) = (A2) q(e, 1, 0) = e and similarly q(e, 1, e) = q(e, 1, q(e, 1, 0)) = (A2) q(e, 1, 0) = e. (2) (1). Let (θ e, θ e ) be the pair of complementary factor congruences associated with e. From the hypothesis and the definition of θ e, θ e it follows that eθ e 0 and 1 θ e e. Since θ e, θ e are congruences, then 0θ e e and e θ e 1. It follows that q(e, 0, e) = 0 and q(e, e, 1) = 1. Then we conclude the proof as follows: q(e, 1, 0) = q(e, q(e, e, 1), q(e, 0, e)) = (A2) q(e, e, e) = (A1) e. Proposition 10. In the hypothesis of the above proposition we have: q(e, e, 0) = e iff θ e = θ(e, 0); q(e, 1, e) = e iff θ e = θ(e, 1). Proof. If q(e, e, 0) = e, then θ(e, 0) {(x, y) : q(e, x, y) = x} = θ e. Conversely, if q(e, x, y) = x, then, by e θ(e, 0) 0 we have x = q(e, x, y) θ(e, 0) q(0, x, y) = y. The second equivalence can be proven in a similar way. Since RCAs are one-pointed algebras, we introduce semicentral elements as the best approximation to central elements of Church algebras. Definition 11. A factor element e of an RCA A is called semicentral if it is idempotent; that is, it satisfies the following further axiom: (A4) q(e, e, 0) = e. We denote by S(A) the set of semicentral elements of A. If e is semicentral, then the pair (θ e, θ e ) of complementary factor congruences determined by the decomposition operation q(e,, ) has a good behaviour. Proposition 11. Let A be an RCA and e A be semicentral. Then, (i) θ e = θ(e, 0), the minimal congruence relating e and 0. (ii) The congruence θ e permutes with every congruence, i.e., φ θ e = θ e φ for every congruence φ Con(A). (iii) a, b A, aθ e q(e, b, a)θ e b. Proof. (i) By the proof of Proposition 10. (ii) Let aφbθ e c for some b. By aφb we have q(e, c, a)φq(e, c, b) and by bθ e c we have q(e, c, b) = c. Moreover, aθ e q(e, c, a) holds because q(e, a, q(e, c, a)) = (A2) q(e, a, a) = (A1) a. In conclusion, aθ e q(e, c, a)φc.

12 12 K. Cvetko-Vah and A. Salibra Algebra univers. (iii) Since (θ e, θ e ) is a pair of complementary factor congruences, then there exists a unique u such that bθ e uθ e a. Notice that 0θ e e follows from q(e, e, 0) = e. Then we have: a = q(0, b, a)θ e q(e, b, a)θ e q(e, b, u) = b, where the conclusion q(e, b, u) = b follows from the assumption bθ e u. In the following proposition we show that q(x, x, 0) is semicentral for every factor element x. Proposition 12. Let A be an RCA. Then (S(A), q, 0) is a retract of (F(A), q, 0) through the onto homomorphism f : F(A) S(A) defined by f(x) = q(x, x, 0). Proof. Let x, y, z be factor elements. We show that f is a homomorphism: q(f(x), f(y), f(z)) = q(f(x), q(x, f(y), f(y)), q(x, f(z), f(z))) by (A1) = q(x, q(x, q(y, y, 0), q(z, z, 0)), q(z, z, 0)) by (A3) = q(x, q(y, y, 0), q(z, z, 0)) by (A2) = q(q(x, y, z), q(x, y, z), 0) by (A3) = f(q(x, y, z)) by def. f By Proposition 7 q(x, x, 0) is a factor element if x is such. We now show that q(x, x, 0) is idempotent: q(q(x, x, 0), q(x, x, 0), 0) = q(x, q(x, x, 0), 0) by (A3), (A1) = q(x, x, 0) by (A2) 4. Semicentral RCAs In a generic RCA there is no need for the set of semicentral elements to comprise all of the algebra. In this section, we define under the name of semicentral RCA those RCAs where this actually happens. We prove that every variety of semicentral RCAs is term equivalent to a variety of right-handed SBAs with additional operations. Definition 12. An RCA is called a semicentral RCA ( SRCA, for short) if every element is semicentral. An SRCA is thus a pointed algebra A of type τ for which a ternary term operation q is defined such that q(0, x, y) = y. q(x, x, 0) = x. q(x, y, y) = y. q(w, q(w, x, y), z) = q(w, x, z) = q(w, x, q(w, y, z)). w, q(w,, ) : A A A is a homomorphism w.r.t. the τ-operations. Given an RCA A, by Proposition 12 the algebra (S(A), q, 0) of all semicentral elements of A is an SRCA. The set S(A) is not in general closed under the operations of type τ.

13 Vol. 00, XX The connection of skew Boolean algebras to Church algebras 13 Example 2. Every algebra in a one-pointed discriminator variety is an SRCA with respect to the ternary term operation q(x, y, z) = s(x, 0, z, y). Example 3. Let P f (X) be the set of all finite subsets of a set X. Define a ternary operation on P f (X) as follows: q P (F, G, H) = (G F ) (H \ F ), for all F, G, H f X. Then the algebra (P f (X), q P, ) is an SRCA. Example 4. (see [7]) Let F(X, Y ) be the set of all finite functions from X into Y. If f F(X, Y ) then dom(f) is a finite subset of X. Then the algebra (F(X, Y ), q F, 0) is an SRCA, where 0 = is the empty function; For all finite functions f : F Y, g : G Y and h : H Y (F, G, H f X), q F (f, g, h) = g G F h H\F. The following characterisation of the directly indecomposable SRCAs will be useful in the proof of the main theorem of Section 4.1. Lemma 6. Let A be an RCA. Then A is a directly indecomposable SRCA if, and only if, for all x, y, z A: { y if x 0; q(x, y, z) = z if x = 0. Proof. ( ) Given the assumption on A, for each x A, either θ x = and θ x = or θ x = and θ x =. If x 0, then θ x = θ(x, 0) = {(y, z) : q(x, y, z) = y} =, making q(x, y, z) = y for all y, z. ( ) A is an SRCA because, for every x 0, the condition yz(q(x, y, z) = y) implies that x is semicentral. By the way of contradiction, assume now A to be directly decomposable. Let (φ, φ) be a pair of nontrivial complementary factor congruences, and let a 0. Since φ φ =, then there exists b A such that 0φb φa. We have two cases. (b = 0): from 0 φa it follows that y = q(0, x, y) φq(a, x, y) = x, contradicting the nontriviality of φ. (b 0): from 0φb it follows that y = q(0, x, y)φq(b, x, y) = x, contradicting the nontriviality of φ. In particular, the rule for q in Lemma 6 holds in every subdirectly irreducible SRCA A term equivalence result. In this section we prove that every variety of SRCAs of type τ is term equivalent to a variety of right-handed SBAs with additional regular operations.

14 14 K. Cvetko-Vah and A. Salibra Algebra univers. We start with the pure case, where τ = {q, 0}. Consider the following correspondence between the algebraic similarity types of SBAs and of pure SRCAs. Beginning on the SBA side: q(x, y, z) := (x y) (z \ x) and 0 := 0 Beginning on the SRCA side: x y := q(x, x, y), x y := q(x, y, 0) y \ x := q(x, 0, y) and again 0 := 0. If B is an SBA, then B q = (B; q, 0) denotes the corresponding algebra in the similarity type of SRCAs. Similarly, if A is an SRCA, then A = (A;,, \, 0) denotes the corresponding algebra in the similarity type of SBAs. Theorem 2. The above correspondences define a term equivalence between the varieties of pure SRCAs and of right-handed SBAs. More precisely, (i) If A is a pure SRCA, then A is a right-handed SBA; (ii) If B is a right-handed SBA, then B q is a pure SRCA; (iii) (A ) q = A and (B q ) = B. Proof. Given a subdirectly irreducible SRCA, by Lemma 6 we have: { x if x 0; x y = q(x, x, y) = y if x = 0. { y if x 0; x y = q(x, y, 0) = 0 if x = 0. { 0 if x 0; y \ x = q(x, 0, y) = y if x = 0. But these are primitive right-handed SBA operations. Thus the derived algebra of a subdirectly irreducible SRCA satisfies all identities holding for all righthanded SBAs. Since any SRCA A is a subdirect product of irreducible cases, the derived algebra A = (A;,, \, 0) is a right-handed SBA. Conversely, if B is a primitive right-handed SBA, setting q(x, y, z) = (x y) (z \ x), one has { y if x 0; q(x, y, z) = z if x = 0. Thus (B, q, 0) is a directly indecomposable SRCA by Lemma 6. Hence, for any right-handed SBA B, B q = (B, q, 0) is an SRCA since right-handed SBAs are subdirect products of right-handed primitive SBAs. That (A ) q = A and (B q ) = B are easily seen when A and B are subdirectly irreducible. Therefore this occurs in general. Remark 1. Directly indecomposable (subdirectly irreducible) SRCAs thus correspond to directly indecomposable (subdirectly irreducible) right-handed SBAs. In particular, the rule for q(x, y, z) in Lemma 6 provides directly indecomposable SRCAs. The derived SRCA (3 R ) q corresponds to the subdirectly irreducible

15 Vol. 00, XX The connection of skew Boolean algebras to Church algebras 15 right-handed SBA 3 R defined in Section 2.4. As such, the subdirectly irreducible algebra (3 R ) q is a generator for the variety of all pure SRCAs. Also, an SRCA equation is an identity for all pure SRCAs iff that equation is an identity in (3 R ) q. Corollary 1. Let A be an RCA. Then the algebra (S(A),,, \, 0) is a righthanded SBA. Example 5. The reduct (F(X, Y ),,, \, 0) of the SRCA (F(X, Y ), q F, 0) of Example 4 is a right-handed SBA (see also [7]), where f g = f g G\F, f g = g G F and f \ g = f F \G. Theorem 2 can be likewise extended to a generic variety of SRCAs of type τ. Definition 13. An algebra of type τ is a right-handed SBA with additional regular operations if it is a right-handed SBA satisfying the following identity for all g τ: (x g(ȳ)) (g( z) \ x) = g((x y 1 ) (z 1 \ x),..., (x y n ) (z n \ x)). Theorem 3. Every variety of SRCAs of type τ is term equivalent to a variety of right-handed SBAs with additional regular operations SRCAs and the Green s relation D. Let A be an SRCA. Define the relation D on A as follows: xdy (y x = x) and (x y = y). Besides the equivalent conditions of Lemma 5 for right-handed SBAs we now have: Proposition 13. For any SRCA A the following conditions are equivalent, for all x, y A: (1) xdy; (2) θ x = θ y ; (3) yθ x 0 and xθ y 0; (4) q(x, a, b) = q(y, a, b) for all a, b A. Proof. (2) (3) Recall that θ x = θ(x, 0) and that yθ x 0 is equivalent to θ(y, 0) θ(x, 0). (3) (1) yθ x 0 iff q(x, y, 0) = y. Similarly for xθ y 0. (1) (4) Given xdy, by applying the term equivalence of Theorem 2 we have: q(x, a, b) = (x a) (b \ x) = Lemma 5 (y a) (b \ y) = q(y, a, b). (4) (1) x y = q(x, y, 0) = q(y, y, 0) = y and y x = q(y, x, 0) = q(x, x, 0) = x. Proposition 14. Let A be an SRCA. Then the following conditions hold: (1) D is a congruence on the pure reduct A 0.

16 16 K. Cvetko-Vah and A. Salibra Algebra univers. (2) On each equivalence class D x (x 0), q(x, y, z) = y for all y, z D x. Proof. (1) The pure SRCA A 0 and the right-handed SBA (A 0 ) share a common underlying set, with the operations of each algebra polynomiallydefined from those of the other. Thus both algebras share common congruences, making the congruence D on (A 0 ) a congruence on A 0. (2) By Lemma 5 x y = y = y x and x \ y = 0 = y \ x in all D-classes of a right-handed SBA. Then, by the term equivalence result of Theorem 2 in each D-class we have: q(x, y, z) = (x y) (z \ x) = y 0 = y. Example 6. Let (F(X, Y ), q F, 0) be the algebra of Example 4. If f : F Y and g : G Y are finite functions (F, G f X), then we have that fdg iff F = G. The pair of complementary factor congruences associated with f : F Y is defined as follows: g θ f h iff g G F = h H F ; g θ f h iff g G\F = h H\F. Then the algebra (F(X, Y ), q F, 0) can be decomposed as a Cartesian product of (F(F, Y ), q F, 0) and the algebra (F(X \ F, Y ), q F, 0): (g : G Y ) (g G F, g G\F ) Ideals. The bijective correspondence between the Boolean algebra of central elements and the Boolean algebra of factor congruences is one of the most important properties of Church algebras. RCAs lack this full correspondence. Semicentral elements of an RCA correspond to a proper subset of the set of factor congruences. In this section we introduce ideals of RCAs and show that, for every semicentral element e, the equivalence class 0/θ e is a principal ideal, which is an idempotent semi-boolean-like-algebra (see [16, Definition 4.1]). We also characterise the Boolean algebra of central elements of a principal ideal. Definition 14. Let A be an RCA. An ideal I is a subalgebra of the pure reduct A 0 satisfying the following condition: x A and y, z I q(x, y, z) I. Notice that, if I is an ideal, then x y, y \ x I for all x A and y I. An ideal I is principal if there is an element e I such that q(e, x, y) = x for all x, y I. In such a case, the principal ideal I is said to be generated by e. Proposition 15. Let A be an RCA and e A be semicentral. Then we have: (1) The equivalence class 0/θ e is a principal ideal generated by e. (2) The algebra A e = (0/θ e, q Ae, 1 Ae, 0 Ae ), where q Ae = q A, 1 Ae = e and 0 Ae = 0 A, is a Church algebra.

17 Vol. 00, XX The connection of skew Boolean algebras to Church algebras 17 Proof. (1) 0/θ e is an ideal because q(x, y, z)θ e q(x, 0, 0) = 0 for all x A and y, z 0/θ e. By q(e, e, 0) = e we get e 0/θ e. Moreover, the ideal 0/θ e is generated by e since q(e, x, y) = x for all xθ e y. (2) By definition of θ e we have xθ e 0 iff q(e, x, 0) = x. We recall from [16] that a double-pointed algebra B is a semi-boolean-likealgebra if it is a Church algebra all of whose elements are factor elements (that is, they satisfy axioms (A1)-(A3) of Proposition 6). B is idempotent if it satisfies (A4). By Theorem 2 and Lemma 2 the relation, defined by is a partial ordering in every SRCA. x y iff x y = x = y x, Proposition 16. Let A be a pure SRCA and e A. Then we have: (1) The Church algebra A e is an idempotent semi-boolean-like-algebra. (2) e = {x : x e} is a subalgebra of A e. (3) The algebra ( e,,,, e, 0), where x y = q(x, e, y); x y = q(x, y, 0); x = q(x, 0, e), is the Boolean algebra of central elements of A e. Proof. (1) The identities (A1)-(A4) hold in A e because they hold in the algebra A. (2) We have e 0/θ e. Let x, y, z e. We show that q(x, y, z) e: q(x, y, z) e = q(q(x, y, z), e, 0) by Def. = q(x, q(y, e, 0), q(z, e, 0)) by (A3) = q(x, y, z) by y, z e The other identity e q(x, y, z) = q(x, y, z) holds because q(x, y, z) 0/θ e. (3) We recall from [16] that an element x of a semi-boolean-like algebra is central if, and only if, q(x, 1, 0) = x. Since 1 = e, then an element x 0/θ e (that is, q(e, x, 0) = x) is central iff q(x, e, 0) = x. It follows that x 0/θ e is central iff x e A generalised Boolean algebra of factor congruences. Factor congruences in a general algebra do not satisfy any particular condition. For example, the set of factor congruences is not in general a sublattice of the lattice of all congruences. In this section we show that, if A is an RCA and Con F (A) = {θ e : e S(A)}, then a structure of generalised Boolean algebra can be given to the set Con F (A). Consider the algebra (Con F (A),,, \, 0), whose operations are defined as follows, for every φ, ψ Con F (A): (i): 0 =, the diagonal relation; (ii): φ ψ = φ ψ

18 18 K. Cvetko-Vah and A. Salibra Algebra univers. (iii): φ ψ = φ ψ, where the symbol denotes the composition of binary relations. (iv): φ \ ψ = φ ψ, where ψ is the factor congruence which is the complement of ψ. Recall that the join of two factor congruences is just the composition of the relations. Proposition 17. The function θ : S(A) Con F (A), mapping e θ e, is a homomorphism from the right-handed skew Boolean algebra (S(A),,, \, 0) onto the algebra (Con F (A),,, \, 0). The kernel of the map θ is Green s relation D. Proof. (1) θ e d = θ e θ d : If q(e, x, y) = x and q(d, x, y) = x then q(q(e, d, 0), x, y) = q(e, q(d, x, y), y) by (A3) = q(e, x, y) by q(d, x, y) = x = x by q(e, x, y) = x If q(q(e, d, 0), x, y) = x, then q(e, x, y) = q(e, q(q(e, d, 0), x, y), y) by q(q(e, d, 0), x, y) = x = q(e, q(e, q(d, x, y), y), y) by (A3) ( ) = q(e, q(d, x, y), y) by (A2) = q(q(e, d, 0), x, y) by (A3) = x By (*) we have x = q(e, x, y) and x = q(e, q(d, x, y), y). We conclude as follows: x = q(e, q(d, x, y), y) = (A3) q(d, q(e, x, y), y) = q(d, x, y). (2) θ e d = θ e θ d : Let x θ e y θ d z. Then we have: q(q(e, e, d), x, z) = q(q(e, e, d), q(e, x, y), z) by q(e, x, y) = x = q(e, q(e, x, z), q(d, y, z)) by (A3) = q(e, q(e, x, z), y) by q(d, y, z) = y = q(e, x, y) by (A2) = x by q(e, x, y) = x In the opposite direction, let q(q(e, e, d), x, z) = x. x θ e q(e, z, x) θ d z. We have Then we prove that z = q(q(e, e, d), z, x) by q(q(e, e, d), z, x) = z = q(e, q(e, z, x), q(d, z, x)) by (A3) = q(e, z, q(d, z, x)) by (A2) = q(d, z, q(e, z, x)) by (A3) Moreover, q(e, x, q(e, z, x)) = x by (A2).

19 Vol. 00, XX The connection of skew Boolean algebras to Church algebras 19 (3) θ e\d = θ e θ d : Let q(e, x, y) = x and q(d, x, y) = y. Then q(q(d, 0, e), x, y) = q(q(d, 0, e), x, q(d, x, y)) by q(d, x, y) = y = q(d, x, q(e, x, y)) by (A3) = q(d, x, x) by q(e, x, y) = x = x by (A1) In the opposite direction, let q(q(d, 0, e), x, y) = x, so that q(d, y, q(e, x, y)) = x. Then we have: and q(d, x, y) = q(d, q(d, y, q(e, x, y)), y) by q(d, y, q(e, x, y)) = x = q(d, y, y) by (A2) = y by (A1) q(e, x, y) = q(e, q(d, y, x), y) by q(d, x, y) = y = q(d, y, q(e, x, y)) by (A3) = x by q(d, y, q(e, x, y)) = x By Proposition 13 the kernel of the map θ is Green s relation D. Corollary 2. The algebra (Con F (A),,, \, 0) is a generalised Boolean algebra of permuting factor congruences. 5. A characterisation of one-pointed discriminator varieties In this section we characterise pointed discriminator varieties in terms of SRCAs and provide a new system of axioms for skew Boolean -algebras. We recall that a pointed variety V is 0-regular if the congruences of algebras in V are uniquely determined by their 0-classes. Fichtner [8] has shown that a pointed variety is 0-regular if, and only if, there exist binary terms d 1 (x, y),..., d n (x, y) satisfying the following two conditions: (1) d i (x, x) = 0 for every i = 1,..., n; (2) d 1 (x, y) = 0... d n (x, y) = 0 x = y. Theorem 4. The following conditions are equivalent for a pointed variety V of type τ: (1) V is a discriminator variety; (2) V is a variety of SRCAs satisfying the following identities for a suitable term d(x, y): d(x, x) = 0; q(d(x, y), x, y) = x. (3) V is a 0-regular variety of SRCAs. (4) V is a variety of SRCAs satisfying the following identities for a suitable term x y: x x = x; q(x y, y, x) = x.

20 20 K. Cvetko-Vah and A. Salibra Algebra univers. (5) V is a variety of SRCAs satisfying the following identities for a suitable term x y: x x = x; x y = y x; q(x y, x, 0) = x y. (6) V is a variety of right handed skew Boolean -algebras with additional regular operations (see Definition 13). Proof. (1 2) Let A be a subdirectly irreducible algebra in V. Define q(x, y, z) = s(x, 0, z, y). The algebra A is an RCA because q(0, y, z) = s(0, 0, z, y) = z. If x 0 then q(x, y, z) = s(x, 0, z, y) = y; thus A is an SRCA by Lemma 6. Define d(x, y) = s(x, y, 0, s(x, 0, y, x)). Then d(x, x) = s(x, x, 0, s(x, 0, x, x)) = 0. We prove the other identity. Assume x y. q(d(x, y), x, y) = s(s(x, y, 0, s(x, 0, y, x)), 0, y, x) = s(s(x, 0, y, x), 0, y, x) by x y if x = 0: = s(y, 0, y, x) = x if x 0: = s(x, 0, y, x) = x. (2 3) If d(x, y) = 0 then x = q(d(x, y), x, y) = q(0, x, y) = y. (3 1) Let A V be a subdirectly irreducible algebra. Define d(x, y) = d 1 (x, y) d n (x, y). We have d(x, x) = 0 because 0 0 = 0. Let a b. If n = 1, then d(a, b) = d 1 (a, b) 0. If n > 1, then there exists a least 1 i n such that d i (a, b) 0. In the following we write d j for d j (a, b) (j = 1,..., n). Define d(a, b) = (d 1 d i 1 ) d i (d i+1 d n ) = 0 d i (d i+1 d n ) = d i (d i+1 d n ) = q(d i, d i, d i+1 d n ) = d i (by Lemma 6) 0. s(x, y, z, t) = q(d(x, y), t, z). Then we have: s(x, x, z, t) = q(d(x, x), t, z) = q(0, t, z) = z. Let now x y, so that d(x, y) 0. Then by Lemma 6 we obtain s(x, y, z, t) = q(d(x, y), t, z) = t for all t, z A. Then s is the switching quaternary term. (4 2) Define d(x, y) = (x y)\x y. Then we have: d(x, x) = (x x)\x x = x \ x = 0. We now show that the identity q(d(x, y), x, y) = x holds in every subdirectly irreducible member A of the variety. Let x y. Then A = x y 0 since by Lemma 6 { y if x = 0; x y = x if x 0.

21 Vol. 00, XX The connection of skew Boolean algebras to Church algebras 21 It follows that the identity q(x y, x, y) = x holds. In conclusion, we have: q(d(x, y), x, y) = q((x y) \ x y, x, y) by Def. = q(q(x y, 0, x y), x, y) by Def. = q(x y, y, q(x y, x, y)) by (A3) = q(x y, y, x) by above = x by Hyp. (2 4) Define x y = (x y) \ d(x, y). In every subdirectly irreducible member of the variety we have: { x if x = y; x y = q(d(x, y), 0, x y) = 0 if x y. Then the identities x x = x and q(x y, y, x) = x are easily derived. The equivalence of (4), (5) and (6) can be easily derived by showing that in every subdirectly irreducible member of the variety we have: { x if x = y; x y = 0 otherwise. 6. Rings Rings are a good source of examples of skew lattices and SBAs (see [11]). We recall from [6] that, if the idempotents of a ring R are closed under multiplication, the idempotents form an SBA under the operations x y = xy, x y = x+y +yx xyx yxy and x\y = x xyx. More generally, any maximal normal band of idempotents in a ring forms an SBA under the above operations. The given join operation becomes the more familiar join x y = x + y xy in the left or right handed cases. A ring R is an RCA with respect to the operation q(x, y, z) = xy + z xz. In this section we show that the semicentral elements of R are the idempotents e satisfying the identity exy = xey for all x, y R. Theorem 5. Let R be a ring and set q(x, y, z) = xy + z xz. Then: (i) (R, q, 0) is an RCA, (ii) S(R) = {e E(R) x, y R : exy = xey}, where E(R) denotes the set of all the idempotents of R, (iii) S(R) contains the generalised Boolean algebra of all central idempotents of R, and (iv) S(R) is a right-handed SBA for the operations defined by x y = xy, x y = x y = x + y xy and x \ y = x yx. Proof. (i) q(0, y, z) = 0 + z 0 = z. (ii) Note that all elements of R satisfy (A1), and that (A2) and (A4) are satisfied exactly by the idempotents of R. Hence S(R) E(R) and S(R) is

22 22 K. Cvetko-Vah and A. Salibra Algebra univers. the set of all the idempotents e in R that satisfy (A3) for either of the basic operations +,,. First we check that all the idempotents satisfy (A3) for either + or. Indeed, for e E(R) we have and q(e, x 1 + x 2, y 1 + y 2 ) = e(x 1 + x 2 ) + y 1 + y 2 e(y 1 + y 2 ) = ex 1 + y 1 ey 1 + ex 2 + y 2 ey 2 = q(e, x 1, y 1 ) + q(e, x 2, y 2 ), q(e, x, y) = e( x) + ( y) e( y) = ex y + ey = (ex + y ey) = q(e, x, y). Finally, e satisfies (A3) for the basic operation if and only if q(e, x 1 x 2, y 1 y 2 ) = q(e, x 1, y 1 )q(e, x 2, y 2 ) holds for all x 1, x 2, y 1, y 2 R. This further simplifies to ex 1 y 2 ex 1 ey 2 + y 1 ex 2 y 1 ey 2 ey 1 ex 2 + ey 1 ey 2 = 0. (6.1) We claim that e E(R) satisfies condition (6.1) for all x 1, x 2, y 1, y 2 R if and only if it satisfies condition exy = xey, (6.2) for all x, y R. So, assume (6.1) and take x 1 = x 2 = 0, y 2 = e. Then (6.1) simplifies to y 1 e = ey 1 e, for all y 1 R. Using this, (6.1) simplifies to ex 1 y 2 = x 1 ey 2, for all x 1, y 2 R, which is exactly our condition (6.2). To prove the converse, assume that (6.2) holds for all x, y R. Taking y = e we get exe = xe, for all x R, and thus the left side of (6.1) simplifies to ex 1 y 2 x 1 ey 2, which is equal to 0 by (6.2). Thus, (6.1) follows. (iii) Follows from (ii), since condition (2) is satisfied by all central idempotents e. (iv) S(R) forms a right-handed SBA for the operations defined by x y = q(x, y, 0), x y = q(x, x, y) and x \ y = q(y, 0, x). In our case this simplifies to x y = q(x, y, 0) = xy, x y = q(x, x, y) = x + y xy = x y and x \ y = q(y, 0, x) = x yx. Acknowledgment. We hereby acknowledge the contribution by an anonymous referee, whose suggestions helped us to correct numerous inaccuracies in a first draft of this paper. References [1] Bauer, A., Cvetko-Vah, K., Stone duality for skew Boolean algebras with intersections, Houston J. Math., 39, 2013, [2] Bigelow D., Burris S.N., Boolean algebras of factor congruences. Acta Sci. Math., 54:11-20, [3] Bignall R.J., Leech J., Skew Boolean algebras and discriminator varieties, Algebra Universalis, 33, 1995, pp [4] Burris S.N., Sankappanavar H.P., A Course in Universal Algebra, Springer, Berlin, [5] Cvetko-Vah, K., Internal decompositions of skew lattices, Communications in Algebra, 35(1), (2007): pp

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