Some classes of directoid groups

Directoid groups are 2-torsion-free abelian groups with an extra binary operation assigning upper bounds. They thus constitute a generalization of abelian l-groups and an equational substitute for directed abelian groups. We discuss some classes of directoid groups, including varieties.

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Datum:	2009
Hauptverfasser:	Gardner, B.J., Parmenter, M.M.
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Veröffentlicht:	Інститут прикладної математики і механіки НАН України 2009
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spelling	irk-123456789-1533812019-06-15T01:26:32Z Some classes of directoid groups Gardner, B.J. Parmenter, M.M. Directoid groups are 2-torsion-free abelian groups with an extra binary operation assigning upper bounds. They thus constitute a generalization of abelian l-groups and an equational substitute for directed abelian groups. We discuss some classes of directoid groups, including varieties. 2009 Article Some classes of directoid groups / B.J. Gardner, M.M. Parmenter // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 1. — С. 44–58. — Бібліогр.: 17 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 06F20, 08B99. http://dspace.nbuv.gov.ua/handle/123456789/153381 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	Directoid groups are 2-torsion-free abelian groups with an extra binary operation assigning upper bounds. They thus constitute a generalization of abelian l-groups and an equational substitute for directed abelian groups. We discuss some classes of directoid groups, including varieties.
format	Article
author	Gardner, B.J. Parmenter, M.M.
spellingShingle	Gardner, B.J. Parmenter, M.M. Some classes of directoid groups Algebra and Discrete Mathematics
author_facet	Gardner, B.J. Parmenter, M.M.
author_sort	Gardner, B.J.
title	Some classes of directoid groups
title_short	Some classes of directoid groups
title_full	Some classes of directoid groups
title_fullStr	Some classes of directoid groups
title_full_unstemmed	Some classes of directoid groups
title_sort	some classes of directoid groups
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2009
url	http://dspace.nbuv.gov.ua/handle/123456789/153381
citation_txt	Some classes of directoid groups / B.J. Gardner, M.M. Parmenter // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 1. — С. 44–58. — Бібліогр.: 17 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT gardnerbj someclassesofdirectoidgroups AT parmentermm someclassesofdirectoidgroups
first_indexed	2025-07-14T04:36:22Z
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fulltext	Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 1. (2009). pp. 44 – 58 c© Journal “Algebra and Discrete Mathematics” Some classes of directoid groups B. J. Gardner and M. M. Parmenter Communicated by V. V. Kirichenko Abstract. Directoid groups are 2-torsion-free abelian groups with an extra binary operation assigning upper bounds. They thus constitute a generalization of abelian l-groups and an equational substitute for directed abelian groups. We discuss some classes of directoid groups, including varieties. Introduction A directoid is a groupoid satisfying the identities xx ≈ x, xy ≈ yx, (xy)x ≈ xy, x((xy)z) ≈ (xy)z. A directoid group is a 2−torsion-free abelian group with a directoid op- eration · (which we often indicate by juxtaposition) connected with the group addition by the identity x + yz ≈ (x + y)(x + z). The absence of 2-torsion is forced by the other conditions. For this as well as background information and a list of references concerning re- lated topics we refer to [8]. For completeness we note another pertinent reference [3] which appeared subsequent to the submission of [8]. A directoid becomes an up-directed set if we define a ≤ b to mean ab = b and with this notion of order a directoid group becomes a directed abelian group. These processes are reversible: an order of either kind gives rise to a binary operation (generally many such) if ab is defined to 2000 Mathematics Subject Classification: 06F20, 08B99. Key words and phrases: directoid, directed abelian group, variety. Jo u rn al A lg eb ra D is cr et e M at h .B. J. Gardner, M. M. Parmenter 45 be an upper bound of {a, b} subject to some constraints, including the requirement that ab = b if a ≤ b. Again, see [8] for details. Directoid groups can be viewed both as a generalization of abelian l-groups and an attempt at equationalizing directed abelian groups, so comparisons and contrasts between these three structures are of inter- est. Here we study some classes of directoid groups, including varieties. Whereas there are no proper varieties of abelian l-groups, we find several families of varieties of directoid groups and show that the lattice of all varieties is quite complicated. Just how complicated remains to be seen: we make no attempt at an exhaustive listing. There are at least two types of classes which we might informally, though reasonably, regard as “varieties of directed abelian groups”: varieties of directoid groups which contain, along with any member, all its order-isomorphic copies, and, in imitation of the e − varieties of regular semigroups [9], [13] (and some other structures [10]), classes of directed abelian groups closed under order-homomorphic images, products and directed subgroups. We shall see that there are no proper subclasses of the second kind but we present a family of non-trivial varieties of the first kind. Some other rather inter- esting classes of directoid groups, for instance those in which ab = a ∨ b whenever a ∨ b exists, fail to be varieties. We discuss a number of such classes and their interdependence. Our notation follows that of [8] and is generally consistent with that of [1] and [5]. For the reader’s convenience we note two items: ◦ denotes the binary operation in a directoid group given by a ◦ b = −((−a) · (−b)) and ‖ indicates incomparability in any partially ordered system. 1. Non-varieties It is easy to perturb the lattice operation on an l−group to produce a new directoid group - there are examples in [7], while 1.1(iii) of [8] provides an easy procedure for making more. To the extent that directoid groups are seen as a generalization of abelian l-groups, such examples may appear a little artificial, inasmuch as an arbitrary upper bound is substituted for a “natural” one. On the other hand, it does seem worth knowing that if desired, a (2-torsion-free abelian) directed group can be made into a directoid group whose operation coincides with the supremum whenever the latter exists. We shall prove this in 1.2 below. In what follows, a ∨ b, a ∧ b will denote, respectively, the supremum, infimum of a, b whenever they exist. Clearly if a ∨ b exists then so does (−a) ∧ (−b), and then (−a) ∧ (−b) = −(a ∨ b) and thus in a directoid group if a · b = a∨ b then (−a)◦ (−b) = −(a · b) = −(a∨ b) = (−a)∧ (−b), and so on. Jo u rn al A lg eb ra D is cr et e M at h .46 Directoid Groups We define a t.m. directoid group to be a directoid group satisfying the (equivalent) conditions a ∨ b exists ⇒ a · b = a ∨ b; a ∧ b exists ⇒ a ◦ b = a ∧ b, and denote the class of t.m. directoid groups by T M. (t.m. stands for “treillis manqué”.) Lemma 1.1. For elements a, b, c of a partially ordered group we have (a + c) ∨ (b + c) = a ∨ b + c, in the sense that LHS exists if and only if RHS exists and when they exist they are equal. Proof. If a ∨ b exists then a, b ≤ a ∨ b so a + c, b + c ≤ a ∨ b + c. If a + c, b + c ≤ y then a, b ≤ y − c so a ∨ b ≤ y − c and so a ∨ b + c ≤ y. Thus a∨ b + c = (a + c)∨ (b + c). The other part is proved similarly. Theorem 1.2. Every 2-torsion-free abelian directed group has an opera- tion making it a t.m. directoid group. Proof. Let M ⊆ G be as in 1.2 of [8]. If a ∈ M let a · 0 = a ∨ 0 if this exists, otherwise anything suitable and (−a)·0 = a·0−a as usual. Clearly (−a)∨0 exists if and only if a∨0 exists and then (−a)∨0 = a∨0−a. Let · be defined on all of G as in 1.1(iii) of [8]. If d∨ c exists then (d− c)∨ 0 exists and d · c = c + (d − c) · 0 = c + (d − c) ∨ 0 = d ∨ c. A directoid group will be called an m-directoid group if whenever its elements a, b have at least one minimal upper bound, a ·b is minimal. We denote by M the class of m-directoid groups. If c is a minimal upper bound of a, b then for every d, c+d is a minimal upper bound of a+d, b+d and so on so by an argument similar to that used for 1.2 we get Theorem 1.3. Every 2-torsion-free abelian directed group has an opera- tion making it an m-directoid group. A t.m. directoid group is "close to" an l−group. The following have opposite behaviour in a sense. A c.l.directoid group is a directoid group in which a · b = a ∨ b if and only if a and b are comparable. We denote by CL the class of c.l. directoid groups. (c.l. stands for "contralattice".) An example of a c.l. directoid group was given by Jakub́ık [12] (p. 16, Example) but they can be built on any directed group. Theorem 1.4. Every 2-torsion-free abelian directed group has an opera- tion making it a c.l. directoid group. Jo u rn al A lg eb ra D is cr et e M at h .B. J. Gardner, M. M. Parmenter 47 Proof. Let G be a 2-torsion-free abelian directed group, M a set as in 1.2 of [8]. For a ∈ M we define a · 0 = a if a > 0, 2(a∨ 0)if a‖0 but a∨ 0 exists and anything suitable otherwise. If a > 0 then (−a) ·0 = a ·0−a = 0 = (−a)∨ 0. If a ∈ M and a‖0, then −a‖0. If (−a)∨ 0 exists then a∨ 0 exists and thus (−a) · 0 = 0 · a− a = 2(0∨ a)− a = 2(a + (−a)∨ 0)− a = a + 2((−a)∨ 0) whence (−a) · 0− (−a)∨ 0 = a + 2(−a)∨ 0)− (−a)∨ 0 = a+(−a)∨ 0 = 0∨a 6= 0; thus (−a) · 0 6= (−a)∨ 0. If a ∈ M and (−a)∨ 0 doesn’t exist there is no problem. Now if b · c = b ∨ c for some b, c ∈ G then (b − c) ∨ 0 = b ∨ c − c = b · c − c = (b − c) · 0 so b − c and 0 are comparable, whence b and c are comparable. We get some restriction if we specify a condition on the order rather than the directoid operation. A directoid group is a multilattice group (see [16]) if for every upper bound c of elements a, b there is a minimal upper bound m of a, b with m ≤ c. For a nr multilattice group [16] we merely require at least one minimal upper bound for each a, b. (Somewhat cryptically, nr stands for "non relativiste" [16].) We’ll call a directoid group fork-free if its underlying directoid conains no forks [12] i.e. its Hasse diagram does not contain a configuration a · b c {{ {{ {{ {{ { CC CC CC CC C a b . Let FF denote the class of fork-free directoid groups. Applying 1.3 to an nr multilattice group, we get Theorem 1.5. Every 2-torsion-free abelian nr multilattice group has an operation making it a fork-free directoid group. A directed group is an antilattice(Fuchs [6]) if it satisfies (i) a ∨ b exists ⇒ a ≤ b or b ≤ a and (ii) a1, a2 ≤ b1, b2 ⇒ ∃c with a1 ≤ c ≤ b1, a2 ≤ c ≤ b2 (Directed groups satisfying (ii) are called Riesz groups [6].) Every direc- toid group on an antilattice is a c.l. directoid group. But these are also t.m. directoid groups. Conversely, every c.l. directoid group which is also a t.m. directoid group must satisfy (i). If G is a Riesz group and an nr multilattice group (in particular, a multilattice group) then every pair a, b has a minimal upper bound m. If a, b ≤ d then there exists c with a ≤ c ≤ m and b ≤ c ≤ d. By minimality Jo u rn al A lg eb ra D is cr et e M at h .48 Directoid Groups we then have m = c ≤ d. Since d is any upper bound, it follows that m = a ∨ b, so G is an l−group. Hence the fork-free directoid groups which are Riesz groups are precisely the l-groups. Let R denote the class of Riesz directoid groups. We shall now demonstrate that the classes we have been discussing are related by inclusion as shown in the following Hasse diagram, all inclusions being proper. T M RRRRRRRRRRRRRR CL ssssssssss CL ∩ T M M RRRRRRRRRRRRRRR CL ∩M L RRRRRRRRRRRRRRRR CL ∩ L (Note that CL∩L is the class of linearly ordered (directoid) groups.) First, let G be an l-group which is not linearly ordered. Then G, as a directoid group with respect to ∨, is in T M but not in CL. Now let M be a subset of G as in 1.2 of [8]. For a ∈ M and a‖0, let a·0 = 2(a∨0) and as usual (−a) ·0 = a ·0−a = 2(a∨0)−a = a∨0+a∨0−a = a∨0+0∨(−a). Then (−a) · 0 6= (−a) ∨ 0 as a ∨ 0 6= 0. Since a‖0 if and only if −a‖0 we have x ·0 = x∨0 if and only if x and 0 are comparable. If c ·d = c∨d then (c−d) ·0 = c ·d−d = c∨d−d = (c−d)∨0 so c−d and 0 are comparable, i.e. c and d are comparable. Thus G, with ·, is in CL but not T M, so that CL‖T M. LetZ0 denote the group of integers with the discrete order, Z ∗ Z0 the lexicographic product, where Z has its standard order and let (0, a) · (0, 0) = (a, 0) and (0,−a) · (0, 0) = (a,−a) for all positive a. This gives us the directoid group Z ∗ Z0 of 3.2 in [8]. If (m, n)‖(k, l) then m = k and n 6= l. Thus each (m + 1, p) is a minimal upper bound, but there is no supremum. It follows that Z ∗Z0 ∈ CL ∩ T M. But since, e.g., (0, 2) · (0, 0) = (2, 0) is not a minimal upper bound of (0, 2) and (0, 0), this is not an m-directoid group. Hence CL ∩ M ⊂ CL ∩ T M. Consequently M ⊂ T M. We can make Q ∗ Q0 into an m-directoid group by 1.3. (Here, as in Z in the previous example, we are using the linear and discrete orders on Q.) As before, every pair of incomparable Jo u rn al A lg eb ra D is cr et e M at h .B. J. Gardner, M. M. Parmenter 49 elements must have the form (r, s), (r, u), where s 6= u. Every (r + ǫ, a), where ǫ > 0 and a is arbitrary, is an upper bound for such a pair, so the pair has no minimal upper bounds. Hence Q ∗Q0 is in M, and certainly in CL. But it is not in L, so CL ∩M 6⊆ L. Since CL ∩ L is the class of linearly ordered groups, we have CL ∩ L ⊂ L and L 6⊆ CL ∩M. From the latter we get L‖CL ∩M. Since CL ∩ L ⊂ CL ∩M, we have L ⊂ M. Since L 6⊆ CL ∩M, so M 6= CL ∩M. Since CL∩L is the class of linearly ordered groups, L is not contained in CL ∩ T M, so neither is M. Let us note, finally, that the above discussion (though not the dia- gram) yields the equation R∩ FF = L. In the next section we shall see some examples of varieties of directoid groups. For the present we note Proposition 1.6. None of the classes M, T M, CL,R,FF is a variety of directoid groups. Proof. Jakub́ık [11] has shown that every directoid group is isomorphic to a directoid subgroup of an m- (whence t.m.) directoid group. This accounts for M and T M. If we take the unique directoid structure on Z with its standard order and the l−group product Z × Z, then Z ∈ CL but Z×Z /∈ CL, so CL is not a variety. The Jaffard group J = {(m, n) ∈ Z × Z : m ≡ n (mod 2)} is a directed subgroup of Z×Z. We can make J a directoid group (e.g. as in Example 2.10 of [7]) and then by 2.9 of [8] we can make a directoid group G on Z × Z with J as a directoid subgroup. Now J /∈ R as pairs of its elements can have more than one minimal upper bound. However G is in R. Hence R is not a variety. (Note that R is closed under convex directoid subgroups and by Proposition 2.4 and Proposition 5.3 of Fuchs [6] also closed under direct products and homomorphic images.) Finally, FF is not homomorphically closed. This is conveniently demonstrated by means of the example which follows. Example 1.7. Let A = Z×Z×Z and let (x1, x2, x3) ≤ (y1, y2, y3) mean that x1 = y1, x2 = y2 and x3 = y3 or x1 ≤ y1, x2 ≤ y2 and x3 + 2 ≤ y3 (cf. Example 2.7 of [7]). A little checking will show that we get a set M as in 1.2 of [8] by taking all (a, b, c) such that (0, 0, 0) < (a, b, c) or a ≥ 0, b ≥ 0, c = 1 or a ≥ 0, b ≥ 0, (a, b) 6= (0, 0), c ≤ 0 or a > 0, b < 0. For (a, b, c) ∈ M we define (i) (a, b, c) · (0, 0, 0) = (a, b, c) if (a, b, c) ≥ (0, 0, 0), (ii) (a, b, 1) · (0, 0, 0) = (a, b, 3) if a, b ≥ 0, Jo u rn al A lg eb ra D is cr et e M at h .50 Directoid Groups (iii) (a, b, c) · (0, 0, 0) = (a, b, 2) if a ≥ 0, b ≥ 0, (a, b) 6= (0, 0), c ≤ 0, (iv) (a, b, c) · (0, 0, 0) = (a + 2, 0, 0 ∨ c + 2) if a > 0, b < 0, and then, as usual, (−a,−b,−c) · (0, 0, 0) = (a, b, c) · (0, 0, 0) − (a, b, c). If (r, s, t)‖(0, 0, 0) we have (r, s, t) · (0, 0, 0) = (∗, ∗, t ∨ 0 + 2) and RHS is a minimal upper bound of (r, s, t) and (0, 0, 0). But then if x, y, z ∈ A and x, y ≤ z ≤ x · y, then 0, y − x ≤ z − x ≤ x · y − x = 0 · (y − x) so z − x = x · y − x and hence z = x · y. Thus z is a minimal upper bound, and A is fork-free. Now let B = Z × Z with the product order. We make B a directoid group by defining (a, b) · (0, 0) = { (a, b) if a ≥ 0, b ≥ 0 (a+2,0) if a > 0, b < 0, and (−s) · 0 = s · 0 − s in each case. The function f : A → B with f(a, b, c) = (a, b) for all a, b, c is a surjective directoid group homomor- phism. But if a > 0 and b < 0 then (a, b), (0, 0) ≤ (a + 1, 0) < (a + 2, 0) = (a, b) · (0, 0). We end this section with a further observation about nr multilattice groups. Vaida [16] proved (in a more general setting, without commuta- tivity) that a partially ordered group is an nr multilattice group if and only if it has a generalized Jordan decomposition, i.e. every element t can be written as u−v for some positive elements u, v for which 0 is a maximal lower bound. In the case of 2-torsion-free abelian groups we can prove one implication by a directoid group argument which produces a generalized Jordan decomposition which mimics that in an l−group. We can make an nr multilattice group G into a fork-free directoid group (1.5) and then a ◦ b is always a maximal lower bound of a, b. For every g ∈ G we have (using Proposition 2.6(ii) of [7]) g = g + 0 = g · 0 + g ◦ 0 = g · 0− (−g) · 0 and (g · 0) ◦ (−g) · 0 = 0 by Proposition 2.6(iv) of [7] so 0 is a maximal lower bound of g · 0 and (−g) · 0. 2. Varieties There are no proper subvarieties of abelian l−groups: free abelian l−groups are subdirect products of copies of Z with its linear order [17] and every non-zero abelian l−group clearly contains an isomorphic copy of Z. The situation is considerably more complicated when we come to directoid groups. We shall present a few examples to demonstrate this without attempting a classification. As well, we show the existence of classes with some claim to being viewed as "varieties of directed groups". Jo u rn al A lg eb ra D is cr et e M at h .B. J. Gardner, M. M. Parmenter 51 We shall deal with the following classes. L: The class of abelian l−groups. An (n ∈ Z+, n > 1): The class of directoid groups satisfying (nx · ny) · nz ≈ nx · (ny · nz). Bn (n ∈ Z+, n > 1): The class of directoid groups satisfying nx · ny ≈ n(x · y). In (n ∈ Z, n > 1): The class of n − isolated directoid groups; those satisfying nx ≥ 0 ⇒ x ≥ 0. D: The class of directoid groups satisfying (mx · nx) · kx ≈ mx · (nx · kx) for all m, n, k ∈ Z. (Note that for n = 1, An would be L and Bn and In would be the class of all directoid groups.) The directoid groups in D are called alternating by Kopytov and Dimitrov [14]. With the exception of the In, all of these classes are obviously vari- eties. It follows from the next result (which generalizes Proposition 2.6 (vi) of [7]) that the In are also varieties. Proposition 2.1. The following conditions are equivalent for a directoid group G and a positive integer n. (i) (n + 1)x ≥ 0 ⇒ x ≥ 0. (ii) x · (−nx) ≥ 0 for all x. Proof. (i)⇒(ii). We have x · (−nx) ≥ x, so −x ≥ −(x · (−nx)). Hence x · (−nx) ≥ −nx ≥ −n(x · (−nx)), so (n + 1)(x · (−nx)) ≥ 0, whence x · (−nx) ≥ 0. (ii) ⇒(i). If (n + 1)x ≥ 0 then −(n + 1)x ≤ 0, so 0 = (−(n + 1)x) · 0. But then x = x + 0 = x + (−(n + 1)x) · 0 = (−nx) · x ≥ 0. Corollary 2.2. For n > 0, In+1 is the variety defined by (x · (−nx)) · 0 ≈ x · (−nx). Theorem 2.3. For every integer n > 1 we have Bn ⊂ In, Bn‖An, An‖In. Proof. If a ∈ A ∈ Bn and na ≥ 0, then na = na·0 = n(a·0) so n(a·0−a) = 0. Now a · 0 ≥ a so a · 0− a ≥ 0. Since positive elements can’t have finite order, we have a · 0 = a, i.e. a = a · 0 ≥ 0. Thus A is in In, so Bn ⊆ In. Let m be a positive integer relatively prime to n, Zm 0 the group of integers modulo m with the discrete order, and let H(m) = Z ∗ Zm 0 (lexicographic product) where Z has its standard order (cf. 3.3 of [8]). Let Zm = {0, 1, . . . ,m − 1}, and write H(m) as an internal sum. If for a ∈ Zm, k ∈ Z we have n(k +a) ≥ 0, i.e. nk +na ≥ 0, then either nk = 0 Jo u rn al A lg eb ra D is cr et e M at h .52 Directoid Groups and na = 0 or nk > 0. In the former case, k = 0 and a = 0; in the latter, k > 0. Thus k+a ≥ 0. It follows that with any directoid operation, H(m) is in In. Let rn + sm = 1 (r, s ∈ Z) so that nr = nr = 1. We can make H(m) a directoid group with 1 · 0 = 1. Then nr · n2r = 1 · 2nr = 1 · 2 = 1 + 0 · 1 = 1 + 1 and n(r · 2r) = n(r + 0 · r) = nr + n(0 · r) = 1 + n(0 · r). Now r has finite order and (as nr = 1) it is non-zero, so that r‖0 whence 0 · r > 0. Thus 0 · r = k + a for some a ∈ Zm, k > 0. But then n(r · 2r) = 1 + n(0 · r) = nk + (1 + na) 6= 1 + 1 = nr · n2r, so H(m) /∈ Bn. We have proved that Bn ⊂ In. We next examine An. If m 6= 3, so that 1 6= −21, we can choose our directoid operation in H(m) so that 1 · 0 = 1 and 2 · 0 = 2. Then (nr · 2r) · 0 = (1 · 2) · 0 = (1 + 1 · 0) · 0 = (1 + 1) · 0 = 1 + 1 (as 1 > 0), while nr · (n2r · 0) = 1 · (2 · 0) = 1 · 2 = 2, so H(m) /∈ Am. (For completeness we note that if m = 3 then 21 = −1 so we can make 1 · 0 = 1 and 2 · 0 = 1 · 0− 1 = 1− 1 and then (nr ·n2r) ·0 = (1 ·2) ·0 = (1+0 ·1) ·0 = (1+1) ·0 = 1+1, while nr · (n2r · 0) = 1 · (2 · 0) = 1 · (1 − 1) = 1 − 1 so H(3)(with this directoid operation) /∈ An.) Thus H(m)(with a suitable directoid operation) /∈ An whenever m and n are relatively prime, so that under this condition H(m) ∈ In \ An. As in 3.1 of [8], let Z(n) = Z with u � v if and only if v = u+nc+(n+ 1)d for some c, d ∈ Z+ ∪ {0}. Now if g, h ∈ Z we can assume g ≤ h and then nh−ng = n(h−g) so ng � nh. It follows that � is linear on nZ and for any directoid operation · we have na·nb = na∨nb for all a, b ∈ Z. Thus for all a, b, c ∈ Z we have (na · nb) · nc = (na ∨ nb) · nc = n(a ∨ b) · nc = n(a ∨ b) ∨ nc = (na ∨ nb) ∨ nc = na ∨ (nb ∨ nc) = na ∨ (n(b ∨ c)) = na · (n(b ∨ c)) = na · (nb ∨ nc) = na · (nb · nc), so Z(n) ∈ An (for any directoid operation). On the other hand in Z(n) we have n1 � 0 but 1 6� 0, so Z(n) /∈ In. We thus have Z(n) ∈ An \ In, and this with the result above gives us An‖In for all n. In Z(n) (as n > 1) 2 − 1 can’t be positive, so 1‖2, whence 1 · 2 6= 1, 2 (for any operation ·) and hence n(1 · 2) 6= n, 2n. But n � 2n so n1 · n2 = 2n. Since n(1 · 2) 6= n1 · n2, Z(n) is not in Bn so it is in An \ Bn. Let Z ∗ Z0 be given the directoid group structure used for the discussion of the previous Hasse diagram of classes. If (k, a) ≥ (0, 0) then for n ≥ 2, n(k, a) = (nk, na) ≥ (0, 0) while if (k, a) ≤ (0, 0), then n(k, a) = (nk, na) ≤ (0, 0), as in the first case k > 0 or k = 0 and a = 0 and in the second k < 0 or k = 0 and a = 0. If a ∈ Z+ then n(0, a) · (0, 0) = (0, na) · (0, 0) = (na, 0) = n((a, 0) · (0, 0)) and n(0,−a) · (0, 0) = (0,−na) · (0, 0) = (na,−na) = n(a,−a) = n((0,−a) · (0, 0)). This proves that in Z ∗ Z0 we have nu · 0 = n(u · 0) for all u. If now u, v ∈ Z ∗ Z0 then nu · nv = (nu − nv) · 0 + nv = (n(u − v)) · 0 + nv = n((u − v) · 0) + nv = n((u − v) · 0 + v) = n(u · v). Jo u rn al A lg eb ra D is cr et e M at h .B. J. Gardner, M. M. Parmenter 53 Thus Z ∗ Z0 ∈ Bn for every n ≥ 2. But in the same group we have ((0, n) · (0, 2n)) · (0, 3n) = ((0,−n) · (0, 0) + (0, 2n)) · (0, 3n) = ((n,−n) + (0, 2n)) ·(0, 3n) = (n, n) ·(0, 3n) = (n, n), while (0, n) ·((0, 2n) · (0, 3n)) = (0, n) · ((0,−n) · (0, 0) + (0, 3n)) = (0, n) · ((n,−n) + (0, 3n)) = (0, n) · (n, 2n) = (n, 2n), so Z ∗Z0 does not satisfy (nx · ny) · nz ≈ nx · (ny · nz) and so is in Bn \ An. Thus Bn‖An. Theorem 2.4. For any integer n > 1, (i) the groups in Bn satisfy nx = 0 ⇒ x = 0 and (ii) Bn ∩ An = L ⊂ In ∩ An. Proof. (i) Non-zero elements of finite order are incomparable with 0. If nx = 0 then n(x · 0) = nx · n0 = 0 · 0 = 0, so (as x · 0 ≥ 0) x · 0 = 0. But then x ≤ 0, whence x = 0. (ii) For a, b, c ∈ G ∈ An ∩ Bn we have n((a ·b) ·c) = n(a ·b) ·nc = (na · nb)·nc = na·(nb · nc) = na·n(b · c) = n(a·b · c) so n((a·b)·c−a·(b · c)) = 0, whence (a · b) · c = a · (b · c). Now we consider the group Jn = {(a, b) ∈ Z × Z : a ≡ b(modn)} of 3.4 in [8] with a directoid group structure such that a ·b is the least upper bound of a and b whenever the latter exists in Jn, i.e. we are dealing with a t.m. directoid group. If a, b, c, d ∈ Z then (na, nb), (nc, nd) ∈ Jn and (calculating in the l−group Z×Z) (na, nb)∨(nc, nd) = (na∨nc, nb∨nd) = (n(a∨ c), n(b∨d)). This last is in Jn and is clearly the least upper bound there. Thus we have, in Jn, (na, nb) · (nc, nd) = (n(a ∨ c), n(b ∨ d) = n(a ∨ c, b ∨ d). In particular, nr · ns = n(r · s) for all r, s ∈ Jn. If now r, s, t ∈ Jn, then (nr · ns) · nt = n(r ∨ s) · nt = n((r ∨ s) ∨ t) and nr · (ns · nt) = nr · n(s ∨ t) = n(r ∨ (s ∨ t). This proves that Jn ∈ An. As Jn is a subgroup of an l−group, its order is isolated, (i.e. it satisfies the condition ∀k ∈ Z+, kx ≥ 0 ⇒ x ≥ 0), so certainly Jn ∈ In. (See also 2.7 below.) But Jn is not a lattice as, e.g., (n + 1, 1), (2, n + 2) ∈ Jn, but they have (n+2, n+2) and (2n+1, n+1) as mininal upper bounds. The variety D consists of those directoid groups for which every one- generator directoid subgroup is an l−group. This is asserted without proof in [14]. To prove it we make use of the fact that in any directoid group G ∈ D, if a ∈ G, n, k ∈ Z+ and n > k, then na · 0 ≥ ka and n(a ·0) = (na) ·0. It is shown in [5], p.75, that these are true in l−groups but only the associativity of · on integer multiples of x is needed. Now Z × Z with its standard product l−group structure is a free (abelian) l−group on (−1, 1) [2] and the cited results enable us to define a directoid group homomorphism Z × Z → G under which (−1, 1) 7→ a. This being Jo u rn al A lg eb ra D is cr et e M at h .54 Directoid Groups so for each a, every one-generator directoid subgroup of G is an l−group. The converse is obvious. As noted above, if a ∈ G ∈ D, then n(a ·0) = (na) ·0 for each n ∈ Z+, and we saw in the proof of 2.3 that this property implies G ∈ Bn. We also saw that Z∗Z0, with the operation used earlier, does have the above property and hence is in Bn. However, we also have ((0, 1) · (0, 2)) · (0, 3) = ((0, 1) + (0, 0) · (0, 1)) · (0, 3) = ((0, 1) + (1, 0)) · (0, 3) = (1, 1) · (0, 3) = (1, 1), while (0, 1) · ((0, 2) · (0, 3)) = (0, 1) · ((0, 2) + (0, 0) · (0, 1)) = (0, 1) · ((0, 2) + (1, 0)) = (0, 1) · (1, 2) = (1, 2), so Z ∗ Z0 /∈ D. Hence D ⊂ Bn. It was shown in [14] (5.4 Corollary) that if A, B ∈ D, then the lexico- graphic product A∗B, with (a, b) · (a′, b′) = (a ·a′, b ·b′) if (a, b)‖(a′, b′), is also in D. Hence if we take the natural orders, (Q × Q) ∗ Q is in D. It is not in L: it is noted in [14] that (Z×Z) ∗Z ∈ D \ L but we are going to use the group based on Q for another purpose as well. For completeness we give an example to show that (Q × Q) ∗ Q /∈ L. We have (1, 2, 5) · ((1, 2,−1) · (3,−1,−6)) = (1, 2, 5) · (3, 2,−1) = (3, 2,−1) and ((1, 2, 5) · (1, 2,−1)) · (3,−1,−6) = (1, 2, 5) · (3,−1,−6) = (3, 2, 5). But also (Q × Q) ∗ Q /∈ An for n ≥ 2, since for a, b, c ∈ Q there exist a′, b′, c′ ∈ Z with na′ = a, nb′ = b and nc′ = c, so that if the group were in An we’d have (a · b) · c = (na′ · nb′) · nc′ = na′ · (nb′ · nc′) = a · (b · c). This gives us (taking account of 2.3) Theorem 2.5. For each n ≥ 2 we have L ⊂ D ⊂ Bn;D‖An. If V is a variety of directoid groups, 0 6= a ∈ G ∈ V, then the cyclic group generated by a · 0 is a directed group isomorphic to Z with its standard order. As noted above, this generates L. We therefore have Theorem 2.6. L is the unique atom in the lattice of varieties of directoid groups. By 2.4 (proof) Jn ∈ An ∩ In and Jn /∈ Bn, so Jn /∈ D. Hence D‖In ∩An. Thus the varieties we have discussed are related as shown in the following diagram, all inclusions being proper and all pairs of varieties for which no connection is indicated being incomparable. Jo u rn al A lg eb ra D is cr et e M at h .B. J. Gardner, M. M. Parmenter 55 In IIIIIIIII An �� mmmmmmmmmmmmmm Bn 66 66 66 66 66 66 66 66 In ∩ An D L Bn ∩ An A variety V of directoid groups has some claim to be regarded as a "variety of directed abelian groups" (2-torsion-free ones) if it satisfies (e): If A ∈ V and B is a directoid group order isomorphic to A, then B ∈ V. At this point we should mention the concept of an e − variety (see, e.g., [9],[13]). A regular semigroup can be regarded as a semigroup with an additional unary operation ′ such that a = aa′a and a′aa′ = a′ for all a. The operation ′ has to be chosen in the way we choose a directoid operation on a directed group. An e − variety of regular semigroups is then a class of regular semigroups closed under homomorphic images, products and regular subsemigroups. Being an e-variety is equivalent to satisfying the regular semigroup analogue of our condition (e). We’ll now look at the corresponding notions for directoid groups. Let I = ⋂ n>1 In = {A : a ∈ A, n ∈ Z+&na ≥ 0 ⇒ a ≥ 0}. The directed groups in I are said to be isolated or to have isolated order. Theorem 2.7. The varieties I, In, n = 2, 3, 4, . . . satisfy condition (e) and I is the smallest variety of directoid groups which does so. Proof. The first assertion is clear. Let W be a variety satisfying condition (e). Then L ⊂ W so every directoid group whose underlying directed group is an l−group must be in W also. Let A be in I. Then A is an ordered subgroup of an l−group L. (This is deducible from a result of Lorenzen [15]; a direct proof was given by Dieudonné [4]. See also [1], §4.5.) By 2.9 of [8] the directoid group structure on A extends to one on L(which in general is a perturbation of the original l−group structure). Since the new directoid group on L is in W, so is A. Thus I ⊆ W. Theorem 2.8. Let H be a non-empty class of directed abelian groups closed under homomorphic images (for order homomorphisms), products and directed subgroups. Then H = {0} or H is the class of all directed abelian groups. Jo u rn al A lg eb ra D is cr et e M at h .56 Directoid Groups Proof. We first note that if f is an order homomorphism then for a, b ≤ c we have f(a), f(b) ≤ f(c) so the class of all directed abelian groups is homomorphically closed. Also, if Aλ is directed for each λ ∈ Λ then if (aλ)Λ, (bλ)Λ ∈ ∏ Aλ then for each λ there is a cλ ∈ Aλ such that aλ, bλ ≤ cλ, whence (aλ)Λ, (bλ)Λ ≤ (cλ)Λ. Thus the class of all directed abelian groups satisfies the three properties we are considering. Now let H 6= 0 satisfy the three conditions. Let V(H) denote the class of directoid groups whose underlying ordered groups are in H. If I ⊳ G ∈ V(H) then as a directed group G/I is in H.Thus G/I ∈ V(H). If {Gλ : λ ∈ Λ} ⊆ V(H) then in ∏ Gλ (the directoid group product) the following are equivalent: (xλ)Λ ≤ (yλ)Λ; (xλ)Λ(yλ)Λ = (yλ)Λ; (xλyλ)Λ = (yλ)Λ; xλyλ = yλ∀λ; xλ ≤ yλ∀λ. Thus the order is the product order and so ∏ Gλ as a directed group is in H, whence V(H) is closed under products. If G ∈ V(H) and H is a directoid subgroup of G, then H is a directed subgroup of the directed group G ∈ H so H ∈ H, i.e. H ∈ V(H). Hence V(H) is a variety of directoid groups and clearly it satisfies condition (e) so by 2.7 I ⊆ V(H). Now let D be any directed abelian group, not necessarily 2−torsion- free. Let f : F → D be a surjective group homomorphism for a suitably large free abelian group F . If we give F the discrete order, f becomes an ordered group homomorphism. Now the lexicographic product Z ∗ F ((n, u) ≥ (m, v) if and only if n > m or (n, u) = (m, v)) is directed and is in I with any directoid operation. Hence the directed group Z ∗ F is in H. Now take the lexicographic product Z ∗D ((m, d) ≥ (n, e) if and only if n > m or n = m and d ≥ e). This is a directed group and the function f∗ : Z ∗ F → Z ∗ D; f∗(n, u) = (n, f(u)) is a surjective ordered group homomorphism so that H contains Z ∗ D and hence also D. Thus (albeit trivially) the 2-torsion-free members of the analogues of e-varieties of directed abelian groups form varieties of directoid groups satisfying (e), but unlike the case of regular semigroups, the converse is false, as can be seen from 2.7. The key to this is 2.11 of [8]. Note that if V is a variety of directoid groups, we do not necessarily get another variety (in particular, a variety satisfying condition (e)) by taking all directoid groups order isomorphic to members of V. Jo u rn al A lg eb ra D is cr et e M at h .B. J. Gardner, M. M. Parmenter 57 Theorem 2.9. Let L̃ denote the class of directoid groups whose underly- ing directed groups are l−groups. Then L̃ is homomorphically closed, closed under direct products and convex directoid subgroups; it is not closed under directoid subgroups. Proof. If A ∈ L̃, let f : A → B be a surjective homomorphism of directed groups. Then by Proposition 2.9 of [7], Ker(f) is convex and if x, y ∈ Ker(f),i.e. x−0, y−0 ∈ Ker(f) then x ·y = x ·y−0 ·0 ∈ Ker(f). Since x, y ≤ x ∨ y ≤ x · y we have x∨ y ∈ Ker(f). Thus Ker(f) is an l−ideal. Hence B is an l− group with respect to the quotient order defined by f . But by 2.10 of [8] this is the order of the directoid group B, so B ∈ L̃. Product closure is clear. If D ∈ L̃ and C is a convex directoid subgroup of D then for any c1, c2 ∈ C there exists a d ∈ C with c1, c2 ≤ d and hence c1, c2 ≤ c1 ∨ c2 ≤ d, so c1 ∨ c2 ∈ C (where c1 ∨ c2 is calculated in D). Clearly then c1 ∨ c2 is a least upper bound for c1 and c2 in C, so C ∈ L̃. For the final assertion, take the Jaffard group as a subgroup of Z × Z, observe that Z × Z is in L̃ and argue as in the proof of 2.7. References [1] A.Bigard, K. Keimel and S.Wolfenstein, Groupes et anneaux réticulés, Berlin etc., Springer, 1977. [2] G. Birkhoff, Lattice-ordered groups, Ann. of Math. 43(1942), 298-331. [3] I. Chajda and M.Kolař́ık, Directoids with an antitone involution, Comment. Math. Univ. Carolinae 48(2007), 555-569. [4] J. Dieudonné, Sur la théorie de la divisibilité, Bull. Soc. Math. France 69(1941), 133-144. [5] L. Fuchs, Partially ordered algebraic systems, Oxford-London-New York-Paris, Pergamon Press, 1963. [6] L. Fuchs, Riesz groups, Ann. Scuola Norm. Sup. Pisa 19(1965), 1-34. [7] B.J. Gardner and M.M. Parmenter, Directoids and directed groups, Algebra Univ. 33(1995), 254-273. [8] B.J. Gardner and M.M. Parmenter, Directoid groups, Czechoslovak Math. J. 58(133)(2008), 669-681. [9] T.E. Hall, Identities for existence varieties of regular semigroups, Bull. Austral. Math. Soc. 40(1989), 59-77. [10] C. Herrmann and A. Semenova, Existence varieties of regular rings and comple- mented modular lattices, J. Algebra, 314(2007), 235-251. [11] J. Jakub́ık, Über halbgeordnete Gruppen mit verallgemeinerter Jordanscher Zer- legung, Rev. Roumaine Math. Pures Appl. 9(1964), 187-190. [12] J. Jakub́ık, On directed groups with additional operations, Math. Bohemica 118(1993), 11-17. [13] J. Kaďourek and M.B. Szendrei, A new approach in the theory of orthodox semi- groups, Semigroup Forum 40(1990) 257-296. Jo u rn al A lg eb ra D is cr et e M at h .58 Directoid Groups [14] V.M. Kopytov and Z.I. Dimitrov, On directed groups, Siberian Math. J. 30(1989), 895-902. [15] P. Lorenzen, Abstrakte Begründung der multiplikativen Idealtheorie, Math. Z. 45(1939), 533-553. [16] D. Vaida, Groupes ordonnés dont les éléments admettent une décomposition jor- danienne genéralisée, Rev. Roumaine Math. Pures Appl. 9(1964), 929-948. [17] E.C. Weinberg, Free lattice-ordered abelian groups. II, Math. Ann. 159(1965), 217-222. Contact information B. J. Gardner Discipline of Mathematics, University of Tasmania, Private Bag 37, Hobart, Tas. 7001 Australia E-Mail: gardner@hilbert.maths.utas.edu.au M. M. Parmenter Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL A1C 5S7 Canada E-Mail: mparmen@mun.ca Received by the editors: 27.03.2008 and in final form 01.05.2009.

Some classes of directoid groups

Institution

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