Pure Skew Lattices in Rings

Transcription

1 Semigroup Forum Vol. 68 (24) c 24 Springer-Verlag New York, LLC DOI:.7/s RESEARCH ARTICLE Pure Skew Lattices in Rings Karin Cvetko-Vah Communicated by Boris M. Schein Abstract Given a ring R, let S R be a pure multiplicative band that is closed under the cubic join operation x y = x + y + yx xyx yxy. We show that (S,, ) forms a pure skew lattice if and only if S satisfies the polynomial identity (xy yx) 2 z = z (xy yx) 2. We also examine properties of pure skew lattices in rings.. Introduction In [4] a skew lattice S is defined as a set S endowed with two operations, denoted by and, where the operations and are both idempotent and associative, and they satisfy the absorption laws x (x y) =x, (x y) y = y and their duals. On a skew lattice S a preorder is defined by x y whenever x y x = x or dually y x y = y. Preorder induces an equivalence relation on S defined by x y ifand only ifx y and y x. Relation is a congruence for either of the operations,. The equivalence classes in respect to are called components of S. Let R be a ring. Skew lattices in rings were also introduced in [4]. The operation is defined as multiplication, while there are two natural ways of defining the operation, namely x y = x + y xy, and x y = x + y + yx xyx yxy. Throughout this paper, a set S R is called a skew lattice if it forms a skew lattice for multiplication and operation. Any multiplicative band S in a ring R that is closed under the operation is called a -band. It is an easy observation that -bands satisfy all axioms for skew lattices except perhaps associativity. A semigroup ofidempotents is called a band. Recall that in a band, the Green s equivalence relations can be defined by: xry (xy = y & yx = x), xly (xy = x & yx = y), xhy (xry & xly), D = L R= R L. Note that our congruence relation equals Green s relation D, while xhy implies x = y.

2 Cvetko-Vah 269 In [2] pure bands ofmatrices were introduced. We can generalize this definition to arbitrary rings. A band S R is called a pure band if S/D is totally ordered. A skew lattice S R is called a pure skew lattice ifthe multiplicative reduct (S, ) is a pure band. The main result ofthis paper is the following. Theorem.. A pure -band S in a ring R is a skew lattice if and only if S satisfies the polynomial identity x (yz zy) 2 =(yz zy) 2 x. Historically, skew lattices arise from pseudolattices which were introduced by Schein in [8], and near lattices which were studied by Schweigert in [9] and []. For the definitions and basic notions on semigroups and skew lattices we refer the reader to [3] and [4], respectively. 2. Pure skew lattices The following is a useful property of multiplicative bands in rings. Proposition 2.. If S R is a band, then (x y) x = x (y x) = x + y yxy for all x, y S. Proof. Expanding the left and middle expressions with the aid of x (x y) = x =(y x) x yields x + y yxy. Corollary2.2. If S R is a band, then xyx = x if and only if y x y = y. Proof. Clearly, y x y = x + y xyx equals y ifand only ifxyx = x. Corollary2.3. Any -band S in a ring R is a regular band. That is, xyxzx = xyzx holds in S. Proof. Given x y in S, denote x y x = x + y yxy by x. It is easily seen that both x x and x y in S. But ifsuch an x exists for all pairs x y in S, then S must be regular. (See [6]). For any multiplicative band S in a ring R we also have: Proposition 2.4. If x y in S, then x y reduces to y + yx yxy, y x reduces to y+xy yxy with both being idempotent in R. Moreover, xy, yx x, while x y, y x y, provided x y and y x also lie in S. Proof. Given xyx = x, x y and y x clearly reduce as stated. A simple multiplication verifies that they are idempotent. Clearly y (x y) y = y ; moreover (x y) y (x y) =y (y + yx yxy) =y + yx yxy = x y.thusx y y in S provided x y is in S. The situation for y x is similar. That xy, yx x if x y is well known.

3 27 Cvetko-Vah 3. Associativityof Throughout this section let S R be a pure -band. An example in [] shows that S need not be a skew lattice, i.e. need not be associative. In order to prove Theorem., we first prove the following two propositions: Proposition 3.. and bca = ba. Let a, b, c S be such that a b c. Then acb = ab Proof. To begin, consider a = a b a = a+b bab. Clearly, aa = a a = a. Moreover, a Db. That ba b = b is again clear. On the other hand, a ba = (a + b bab) b (a + b bab) = (a + b bab)(ba + b bab) = aba + ba + b bab ba bab + bab = a + b bab = a. In any band, xyz = xz if x and z are in the same D -class and y lies either in that class or in some higher D -class. Hence, acb = aa cb = aa b = b. Similarly, bca = ba. Proposition 3.2. Let a, b, c S be such that a b c. Then (a c) b = a (c b) and (b c) a = b (c a). Proof. We calculate the following with the aid of the previous proposition. a (c b) = a (c + bc cbc) = c + bc cbc + ca + ba cba (ca + ba cba)(c + bc cbc) = c + bc cbc + ca + ba cba cac cabc + cacbc bac babc + bacbc + cbac + cbabc cbacbc = c + bc cbc + ca + ba cba cac bac + cbac, and (a c) b = (c + ca cac) b = c + ca cac + bc + bca bcac (c + ca cac)(bc + bca bcac) = c + ca cac + bc + ba bac cbc cba + cbac cabc caba + cabac + cabc + caba cabac = c + ca cac + bc + ba bac cbc cba + cbac = a (c b), By a symmetric argument, one sees that b (c a) =(b c) a. Proof of Theorem.. Given a b c as above, we next calculate all cases of -composition where c is not in the middle. Using Proposition 3. we calculate the following: (a b) c = (b + ba bab) c = c + cb + cba cbab cbc cbac + cbabc,

4 Cvetko-Vah 27 a (b c) = a (c + cb cbc) = c + cb cbc + ca + cba cba ca (c + cb cbc) = c + cb cbc + ca cac cab + cabc, (b a) c = (b + ab bab) c = c + cb + cab cbab cbc cabc + cbabc, b (a c) = b (c + ca cac) = c + ca cac + cb + cab cab cb (c + ca cac) = c + ca cac + cb cbc cba + cbac, (c a) b = (c + ac cac) b = c + ac cac + bc + bac bac (c + ac cac) bc = c + ac + bc cac cbc abc + cabc, c (a b) = c (b + ba bab) = c + bc + bac babc cbc cbac + cbabc, (c b) a = (c + bc cbc) a = c + bc cbc + ac + abc abc (c + bc cbc) ac = c + ac + bc cbc cac bac + cbac, and c (b a) = c (b + ab bab) = c + bc + abc babc cbc cabc + cbabc. The above calculations yield the following differences: a (b c) (a b) c = ca cac cab + cabc cba + cbab + cbac cbabc = c (ab + ba a bab)(c ) = c (ab ba) 2 (c ) b (a c) (b a) c = ca cac cba + cbac cab + cbab + cabc cbabc = c (ab ba) 2 (c ), (c a) b c (a b) = ac cac abc + cabc bac + babc + cbac cbabc = (c ) (ab ba) 2 c, and (c b) a c (b a) = ac cac bac + cbac abc + babc + cabc cbabc = (c ) (ab ba) 2 c. It is now clear that the associativity of is equivalent to (ab ba) 2 c = c (ab ba) 2 c = c (ab ba) 2 holding for all a b c in S. But for a b c we also have a (bc cb) 2 = abc + acb abcb acbc = abc + ab ab abc =,

5 272 Cvetko-Vah and similarly: (bc cb) 2 a = b (ac ca) 2 =(ac ca) 2 b =. All possible cases are exhausted. The associativity of is indeed equivalent to (xy yx) 2 z = z (xy yx) 2. Recall that a band S R is called a normal band, ifs satisfies the identity xyzw = xzyw, or the equivalent, but seemingly broader identity, xyzx = xzyx. If S is a normal band which is also a skew lattice, then S is called a normal skew lattice [5]. Corollary3.3. lattice. If S is a pure normal -band in a ring R, then S is a skew Proof. The identity xyzw = xzyw implies that both x (yz zy) 2 and (yz zy) 2 x must reduce to. The conclusion ofcorollary 3.3 holds even ifs is not pure, but in our case this is an immediate consequence oftheorem.. A particular instance of a normal -band is a rectangular band (satisfying xyz = xz ). All rectangular bands are normal. If S is a multiplicative rectangular band in a ring R, then x y = yx, so that S is closed under and thus forms a single component skew lattice. This is not generally true for the quadratic join x y = x+y xy. 4. Distributivityand normality Recall the following definitions from [6]. Definition 4.. A skew lattice S is symmetric iffor any x, y S each of the equalities x y = y x and x y = y x implies the other. Definition 4.2. A skew lattice S is middle distributive ifit satisfies the following distributive laws: x (y z) x =(x y x) (x z x) () and x (y z) x =(x y x) (x z x). (2) Notice that every skew lattice S in a ring R is symmetric. In [] M. Spinks stated his result that in every symmetric skew lattice the middle distributivity laws () and (2) are equivalent. We can say even more if S is a pure skew lattice. Theorem 4.3. Every -band S R satisfies (). If S is also pure, then S also satisfies (2). In particular, every pure skew lattice in a ring is middle distributive.

6 Cvetko-Vah 273 Proof. Since S is a regular band, x (y z) x = x (y + z + zy yzy zyz) x = xyx + xzx + xzxyx xyxzxyx xzxyxzx = (xyx) (xzx), and () follows. In general, for all x, y, z S, x yz x = x+yz yzxyz, while (x y x)(x z x) = (x + y yxy)(x + z zxz) = x + xz xz + yx + yz yzxz yx yxyz + yxyzxz = x + yz yzxz yxyz + yxyzxz. Hence (2) holds precisely when yzxz+yxyz = yzxyz+yxyzxz for all x, y, z S. Applying purity we get: Case, x, y z : yxz + yxyz = yxyz + yxz. Case 2, x, z y : yzxz + yxz = yzxz + yxz. Case 3, y, z x: yz + yz = yz + yz. Hence in all cases, (2) must hold. Corollary4.4. Every -band S R for which is associative is a middle distributive skew lattice. Proof. Since is associative, S is a symmetric skew lattice satisfying (). By Spinks Theorem [], S must satisfy (2). We investigate yet another concept ofdistributivity. We say that a skew lattice S is left meet distributive ifit satisfies the identity (x y) z = xz yz and S is right meet distributive ifit satisfies the identity x (y z) =xy xz. Proposition 4.5. A pure band S is normal if and only if for all a b c in S, abc = ac and cba = ca. Proof. Given a b c, both abca = a = acba and bacb = bab = bcab hold by Proposition 3.. If abc = ac and cba = ca hold, then cabc = cac = cbac follows and S is seen to be normal. Conversely, given cabc = cbac, we obtain both abc = ac and cba = ca by multiplying cabc = cbac by a respectively on the left and on the right and reducing by Proposition 3.. The connection between normality and middle distributivity was established in [5]. For pure -bands we observe the following:

7 274 Cvetko-Vah Theorem 4.6. A pure -band S R satisfies (x y) z = xz yz if and only if cba = ca for all a b c in S. S satisfies x (y z) =xy xz if and only if abc = ac for all a b c in S. Proof. Let a b cin S. We use Propositions 2.4 and 3. to calculate the following: (a b) c = bc + bac babc = ac bc (a c) b = cb = ab cb (b a) c = bc + abc babc = bc ac (b c) a = ca = ba ca (c a) b = cb + ab cab cb ab = cb + ab cbab (c b) a = ca + ba cba ca ba = ba. By examining the two cases where (x y) z = xz yz need not hold, it is clear that the equality holds precisely when cba = ca. The argument that x (y z) =xy xz ifand only ifabc = ac for all a b c in S is similar. Since a band is normal ifand only ifall ofits totally quasiordered sub-bands are normal, we can generalize a known result about skew lattices in rings, and about middle distributive, symmetric skew lattices in general. Corollary4.7. A -band S R is both right and left meet distributive if and only if it is normal. Proof. If S is both left and right meet distributive, then so is every totally quasiordered subalgebra of S. By the theorem, each such subalgebra is normal and hence so is S. Conversely, if S is normal, then Similarly, (a b) c = ac bc. a (b c) = a (b + c + cb bcb cbc) = ab + ac + acb abcb acbc = ab + ac + acab abacab acabac = ab ac. 5. Maximal pure skew lattices of matrices We denote by M n (K) the algebra ofall square matrices ofdimension n over a field K. A skew lattice S M n (K) is called a skew lattice of matrices. By [7] S is triangularizable. We may therefore assume that all matrices in S are upper-triangular with s and s on the diagonal. The components of S are determined by certain patterns ofs on the diagonal. IfS is a pure skew lattice, then the diagonals ofmatrices in S form a totally ordered set, and r (ab) = min {r (a),r(b)} by [2] (r (x) denotes the rank ofa matrix x). Each component of S therefore consists exactly of all matrices in S having equal rank; moreover:

8 Cvetko-Vah 275 Lemma 5.. Let S be a pure band in M n (K). Then x y if and only if r (x) = r (y). Moreover, if S is also a -band, then r (x y) = max {r (x),r(y)}. Proof. Let r (x) r (y). Then r (xy) =r (x), which implies xy x, and therefore x y. Hence x y y y x and r (x y) =r (y x) =r (y). In this section we shall see how Theorem. can be used to produce examples ofpure skew lattices. Every pure skew lattice S is contained in a maximal pure skew lattice S max with respect to inclusion. In fact, if S max is a maximal pure skew lattice, then S S max is a pure skew lattice ifand only if S is closed under both multiplication and. A band S M n (K) is called a unicellular band ifthe invariant subspaces of K n for S form a chain with respect to inclusion. In [2] unicellular bands were introduced and a full atlas of maximal unicellular bands in M 2 (K), M 3 (K) and M 4 (K) up to similarity was listed. We obtain maximal pure bands in M n (K) in a similar way. Doing so, we use the result from [2] that every pure band is similar to a band with the property (S) S, where : S M n (K) denotes a map which maps each matrix s S into its diagonal s s 22 (s) =.... snn It follows from Propositions 3. and 4.5 that every pure band contained in M 2 (K) orm 3 (K) that is closed under yields a skew lattice (since it has at most two nontrivial components). Examples 5.2 and 5.3 give a full atlas of maximal pure skew lattices in M 2 (K) and M 3 (K) up to similarity. All letters denoting entries ofmatrices are arbitrary elements ofk, and so is. Example 5.2. Up to similarity, maximal pure skew lattices in M 2 (K) are: S 2 = S 22 = { (, { (, ) },I, ) },I. Example 5.3. Up to similarity, maximal pure bands in M 3 (K) are: S 3 =, I S 32 =, I

9 276 Cvetko-Vah S 33 =, S 34 =, x xy y S 35 =, S 36 =, S 37 =, S 38 =, x xy y I I I I I I. Bands S 3, S 33, S 34, S 35, S 36 and S 38 yield maximal skew lattices, while S 32 and S 37 contain one maximal pure skew lattice each. Band S 32 contains maximal pure skew lattice S 32 =, which is obtained in the following way. For a = x x 2 b= y I we obtain ab = x y + x 2 and ba = a. Therefore: a b = b + ba bab = b + a ab = x x y y

10 Cvetko-Vah 277 and b a = b. From a b b follows x =, and S 32 is a maximal pure skew lattice. Similarly, band S 37 contains maximal pure skew lattice S 37 =, I. To find maximal pure skew lattices in M 4 (K), we use Theorem.. Example 5.4. We begin in the same way as in the case M 3 (K). Let us take a maximal pure band B =, x xy xy 2 y y 2 u u 2 u v + u 2 v 2 v v 2 I. In order for a band S B to be associative in, we need to prove that for all x xy xy 2 a= y y 2 b= u u 2 u v + u 2 v 2 v v 2 c= w w 2 w 3 the identity c (ab ba) 2 =(ab ba) 2 c holds. We observe cs = s for all s S, and therefore c (ab ba) 2 =(ab ba) 2. We calculate: (ab ba) 2 = (x u )(v + y v 2 y 2 ) and obtain the condition: (x u )(v + y v 2 y 2 )=. Theorem. yields the following candidates for maximal pure skew lattices:

11 278 Cvetko-Vah S =, S 2 = {,a,b,i}, S 3 = {,a,c,i}, S 4 = {,b,c,i}, S 5 =, S 6 =, x xy y u 2 u 2 v 2 v 2 u 2 u 2 v 2 v 2 c,i c,i, c,i It remains to check which ofthese candidates are closed under multiplication and. We leave it to the reader to show that S, S, S 3 and S 4 are indeed skew lattices, while S 5 contains one maximal pure skew lattice, S 5 =, x xy y, and S 6 also contains one maximal pure skew lattice, namely S 6 =, u 2 u 2 v 2 v 2. I I Acknowledgment The author is very grateful to Professors M. Omladič and J. Leech for their suggestions that led to significant improvements and to my communicating editor for his patience and guidance. References [] Cvetko-Vah, K., Skew lattices of matrices in rings, Preprint. [2] Fillmore, P., G. MacDonald, M. Radjabalipour, H. Radjavi, Towards a classification of maximal unicellular bands, Semigroup Forum 49 (994), ,.

12 Cvetko-Vah 279 [3] Higgins, P. M., Techniques ofsemigroup Theory, Oxford University Press, New York, 992. [4] Leech, J., Skew lattices in rings, A. Universalis 26 (989), [5] Leech, J., Normal skew lattices, Semigroup Forum 44 (992), 8. [6] Leech, J., Recent developments in the theory of skew lattices, Semigroup Forum 52 (996), [7] Radjavi, H., P. Rosenthal, Simultaneous Triangularization, Springer, New York, 2. [8] Schein, B. M., Pseudosemilattices and pseudolattices, Amer. Math. Soc. Transl. 9(2) (983), 6. [9] Schweigert, D., Near lattices, Math. Slovaca 32 (982), [] Schweigert, D., Distributive associative near lattices, Math. Slovaca 35 (985), [] Spinks, M., On middle distributivity for skew lattices, Semigroup Forum 6 (2), Department of Mathematics University of Ljubljana Jadranska 9, Ljubljana, Slovenia karin.cvetko@fmf.uni-lj.si Received April 4, 23 and in final form August 9, 23 Online publication January 2, 24