Lattice groups
In this paper, we introduce some algebraic struc-ture associated with groups and lattices. This structure is a semi-group and it appeared as the result of our new approach to thefuzzy groups andL-fuzzy groups whereLis a lattice. This approachallows us to employ more convenient language of algebraic...
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Kurdachenko, L.A. Yashchuk, V.S. Subbotin, I.Ya. 2019-06-22T12:17:22Z 2019-06-22T12:17:22Z 2015 Lattice groups / L.A. Kurdachenko, V.S. Yashchuk, I.Ya. Subbotin // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 1. — С. 126-141. — Бібліогр.: 6 назв. — англ. 1726-3255 2010 MSC:20M10, 06D99, 20M99. https://nasplib.isofts.kiev.ua/handle/123456789/158004 In this paper, we introduce some algebraic struc-ture associated with groups and lattices. This structure is a semi-group and it appeared as the result of our new approach to thefuzzy groups andL-fuzzy groups whereLis a lattice. This approachallows us to employ more convenient language of algebraic structuresinstead of currently accepted language of functions. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Lattice groups Article published earlier |
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Lattice groups Kurdachenko, L.A. Yashchuk, V.S. Subbotin, I.Ya. |
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Kurdachenko, L.A. Yashchuk, V.S. Subbotin, I.Ya. |
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In this paper, we introduce some algebraic struc-ture associated with groups and lattices. This structure is a semi-group and it appeared as the result of our new approach to thefuzzy groups andL-fuzzy groups whereLis a lattice. This approachallows us to employ more convenient language of algebraic structuresinstead of currently accepted language of functions.
|
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1726-3255 |
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https://nasplib.isofts.kiev.ua/handle/123456789/158004 |
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Lattice groups / L.A. Kurdachenko, V.S. Yashchuk, I.Ya. Subbotin // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 1. — С. 126-141. — Бібліогр.: 6 назв. — англ. |
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AT kurdachenkola latticegroups AT yashchukvs latticegroups AT subbotiniya latticegroups |
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2025-11-26T21:37:23Z |
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2025-11-26T21:37:23Z |
| _version_ |
1850777617047748608 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 20 (2015). Number 1, pp. 126–141
© Journal “Algebra and Discrete Mathematics”
Lattice groups
L. A. Kurdachenko, V. S. Yashchuk, I. Ya. Subbotin
Dedicated to Professor Efim Zelmanov in occasion of his 60th birthday
Abstract. In this paper, we introduce some algebraic struc-
ture associated with groups and lattices. This structure is a semi-
group and it appeared as the result of our new approach to the
fuzzy groups and L-fuzzy groups where L is a lattice. This approach
allows us to employ more convenient language of algebraic structures
instead of currently accepted language of functions.
The purpose of this work is to look with a somewhat different angle
at algebraic structures related to the functions defined on a group. For
every subset M of a set S there exists its characteristic function, that
is the mapping χM : S → {0, 1} such that χM (y) = 1 for all y ∈ M and
χM (y) = 0 for all y /∈ M. In many commonly used cases, a subset of
M is identified with its characteristic function. In 1965, L.A. Zadeh [6]
based on his generalization of the characteristic function introduced the
fuzzy mathematics. Thus, a fuzzy set on a set S is a sort of generalized
“characteristics function” on S, for whose “degrees of membership” we
can use more diverse set than simple {yes, no}. In fact, we can consider
the set L of degrees of membership. In the optimization problems, L may
express the degree of optimality of the choice (in S); in the classification
problems, it may express the degree of membership in a pattern class;
in other contexts other terminologies appear. In fuzzy mathematics, a
habitual step was to review the situation when L = [0, 1] is the usual
closed interval of real numbers with its natural order. The following
2010 MSC: 20M10, 06D99, 20M99.
Key words and phrases: group, lattice, distributive lattice, fuzzy group, semi-
group.
L. A. Kurdachenko, V. S. Yashchuk, I. Ya. Subbotin 127
interpretation justifies this approach: we can consider a value of the
generalized characteristic function as a probability of the fact that the
given element belongs to the given subset. In this way, the algebraic fuzzy
structures were constructed as follows. With every algebraic structure A, a
corresponding fuzzy structure which characterized by a specific functions
of A on [0, 1] associated with this conventional algebraic structure A, was
connected (see, for example, [4]). For instance, in fuzzy group theory the
objects of study are the functions γ : G → [0, 1], G is a group, satisfying
the following conditions:
γ(xy) > γ(x) ∧ γ(y) for all x, y ∈ G; and γ(x−1) > γ(x) for every x ∈ G
(see, for example, [5] S 1.2). Some generalizations have appeared immedi-
ately. More concretely, considerations of the function γ : G → L where L
is a distributive lattice [1] were initiated. The theory of fuzzy groups was
developed quite rapidly. However it was upswing in breadth rather than
depth development. A variety of results obtained there was not planned
properly. Even in the book [5], there were no attempts to systematize
these results. A large array of results on fuzzy groups just has been
collected in this book with no proper arrangement. In the L-fuzzy groups,
regardless of the most common results, there was no serious progress at
all.
Perhaps the key obstacle here is in the interpretation of an algebraic
structure as a function, which is not very convenient most of the time.
Because of that, very often the function γ is interpreted as an all point
function χ(g, γ(g)), g ∈ G. Here χ(g, a) is a function such that χ(g, a)(g) =
a, χ(g, a)(y) = 0 whenever y 6= g. However, in some cases we need to
consider the function γ as an union of all point functions χ(g, a) for
all g ∈ G and a 6 γ(g) (see, for example, [2], [3]). Actually speaking,
the point functions χ(g, a) play here the role of elements, formally the
subfunctions of γ, so that each time it is necessary to implement keep in
mind some special reservations.
In the current article we offer the interpretation of L-fuzzy groups as
sets with operations. With this algebraic approach, the basic concepts
and results of algebraic nature acquire its natural form, and the process of
their appearance becomes more meaningful. We present the basic concepts
of the theory of L-fuzzy groups, as well as the results in the form in which
they are needed to be for our transformation. The resulting structure is
formally different, and therefore the term for it to be used is different. In
the article we are concerned only with the basic concepts, but nevertheless,
our approach will make it possible to see the general structural picture.
128 Lattice groups
As for the term L-fuzzy group, it seems it does not reflect the essence of
the case, so we will use the term group function. We do not seek maximize
generality, it seems more natural to consider the case, when lattice L is
distributive and finite, although the obtained results can be extended on
the case of an arbitrary complete distributive lattice.
Let L be a lattice and G be a group. To avoid misunderstandings, the
identity element of G is denoted by e. We will consider a set LG of all
functions λ : G → L. On this set we define the operations ∧ and ∨ by the
following rules: if λ, µ ∈ LG, then put
(λ∧µ)(x) = λ(x)∧µ(x) and (λ∨µ)(x) = λ(x)∨µ(x) for each x ∈ G.
Clearly the operations ∧ and ∨ are commutative and associative,
(λ ∧ (λ ∨ µ))(x) = λ(x) ∧ (λ ∨ µ)(x) = λ(x) ∧ (λ(x) ∨ µ(x)) = λ(x)
and
(λ ∨ (λ ∧ µ))(x) = λ(x) ∨ (λ ∧ µ)(x) = λ(x) ∨ (λ(x) ∧ µ(x)) = λ(x),
so that λ ∧ (λ ∨ µ) = λ and λ ∨ (λ ∧ µ) = λ. Clearly λ ∧ λ = λ and
λ ∨ λ = λ. Hence a set LG is a lattice.
If a, b ∈ L, then a∨b = b is equivalent to a 6 b. Therefore we can define
an order on LG: for λ, µ ∈ LG will put λ 6 µ if and only if λ(x) 6 µ(x)
for each element x ∈ G.
Suppose now that a lattice L is distributive and finite. Being finite, it
has the greatest element m and the least element 0. For every function
f : G → L define Supp(f) as a subset of all elements x ∈ G such that
f(x) 6= 0.
Let Y be a subset of G and a ∈ L. We define the function χ(Y, a) as
follows:
χ(Y, a) =
{
a, if x ∈ Y
0, if x /∈ Y .
If Y = {y}, then instead of χ({y}, a) we will write χ(y, a). The function
χ(y, a) is called the point function or shorter the point. By its definition,
χ(y, a) ∈ LG. Furthermore, let f ∈ LG. If Supp(f) = {g1, . . . , gn} is finite
and f(gj) = aj, 1 6 j 6 n, then clearly f = χ(g1, a1) ∨ . . . ∨ χ(gn, an).
Define now the binary operation ⊙ on LG by the following rule. Let
µ, ν ∈ LG, and x be an arbitrary element of a group G. Consider the
subset of the lattice L
{µ(y) ∧ ν(z)|u, v are the elements of G such that yz = x}.
L. A. Kurdachenko, V. S. Yashchuk, I. Ya. Subbotin 129
Since L is finite, this subset really is finite. Therefore we can define about
its least upper bound. Put
(µ ⊙ ν)(x) = ∨y,z∈G,yz=x(µ(y) ∧ ν(z)).
We remark that
(µ ⊙ ν)(x) = ∨y∈G(µ(y) ∧ ν(y−1x)) = ∨z∈G(µ(xz−1) ∧ ν(z)).
Consider now some basic properties of this product.
Proposition 1. The following assertions hold:
(i) The operation ⊙ is associative.
(ii) The function χ(e,m) is an identity element of the operation ⊙.
(iii) λ ⊙ (µ ∨ ν) = (λ ⊙ µ) ∨ (λ ⊙ ν) and (µ ∨ ν) ⊙ λ = (µ ⊙ λ) ∨ (ν ⊙ λ)
for all functions λ, µ, ν ∈ LG.
(iv) If x, y ∈ G, a ∈ L, then (χ(y, a)⊙λ)(x) = a∧λ(y−1x); in particular,
if a = ∨x∈Gλ(x), then ((χ(y, a) ⊙ λ)(x) = λ(y−1x).
(v) (λ ⊙ (χ(y, a)))(x) = a ∧ λ(xy−1); in particular, if a = ∨x∈Gλ(x),
then (λ ⊙ χ(y, a))(x) = λ(xy−1).
(vi) if x, y, u ∈ G, a, b ∈ L then (χ(y, a) ⊙ χ(u, b))(yu) = a ∧ b and
(χ(y, a) ⊙ χ(u, b))(x) = 0 if x 6= yu. In other words, χ(y, a) ⊙
χ(u, b) = χ(yu, a ∧ b); in particular, χ(y, a) ⊙ χ(u, a) = χ(yu, a).
(vii) (χ(x, a) ⊙ λ ⊙ χ(x−1, a))(y) = a ∧ λ(x−1yx).
Proof. (i) Let λ, µ, ν ∈ LG. Put κ = λ ⊙ µ and η = µ ⊙ ν. We have
((λ ⊙ µ) ⊙ ν)(x) = (κ ⊙ ν)(x) = ∨y,z∈G,yz=x(κ(y) ∧ ν(z))
= ∨y,z∈G,yz=x(∨u,v∈G,uv=y(λ(u) ∧ µ(v)) ∧ ν(z))
= ∨u,v,z∈G,uvz=x((λ(u) ∧ µ(v)) ∧ ν(z)).
(λ ⊙ (µ ⊙ ν))(x) = (λ ⊙ η)(x) = ∨u,y∈G,uy=x(λ(u) ∧ η(y))
= ∨u,y∈G,uy=x(λ(u) ∧ (∨v,z∈G,vz=y(µ(v) ∧ ν(z))))
= ∨u,v,z∈G,uvz=x(λ(u) ∧ (µ(v) ∧ ν(z))).
Since (λ(u) ∧ µ(v)) ∧ ν(z) = λ(u) ∧ (µ(v) ∧ ν(z)) for all u, v, z ∈ G,
((λ ⊙ µ) ⊙ ν)(x) = (λ ⊙ (µ ⊙ ν))(x)
130 Lattice groups
for each x ∈ G. It implies that (λ ⊙ µ) ⊙ ν = λ ⊙ (µ ⊙ ν).
(ii) Let λ ∈ LG and consider the product λ⊙χ(e,m). By its definition,
(χ(e,m))(e) = m and (χ(e,m))(x) = 0 whenever x 6= 1. We have now
λ(x) ∧ (χ(e,m))(e) = λ(x) ∧ m = λ(x)
and λ(y) ∧ (χ(e,m))(z) = 0 if z 6= 1,
so that
(λ ⊙ χ(e,m))(e) = ∨y,z∈G,yz=1(λ(y) ∧ χ(e,m)(z))
= λ(e) ∧ χ(e,m)(e) = λ(e),
(λ ⊙ χ(e,m))(x) = ∨y,z∈G,yz=x(λ(y) ∧ χ(e,m)(z))
= λ(x) ∧ χ(e,m)(e) = λ(x).
Since it is valid for all x ∈ G, λ ⊙ χ(e,m) = λ. In a similar way we can
prove that χ(e,m) ⊙ λ = λ.
(iii) We have
λ ⊙ (µ ∨ ν)(x) = ∨y∈G(λ(y) ∧ ((µ ∨ ν)(y−1x)))
= ∨y∈G(λ(y) ∧ (µ(y−1x) ∨ ν(y−1x)))
= ∨y∈G(λ(y) ∧ µ(y−1x)) ∨ (λ(y) ∧ ν(y−1x))
= (∨y∈G(λ(y) ∧ µ(y−1x))) ∨ (∨y∈G(λ(y) ∧ ν(y−1x)))
= (λ ⊙ µ)(x) ∨ (λ ⊙ ν)(x)
= ((λ ⊙ µ) ∨ (λ ⊙ ν))(x).
It proves that
λ ⊙ (µ ∨ ν) = (λ ⊙ µ) ∨ (λ ⊙ ν).
Using similar arguments, we obtain that and
(µ ∨ ν) ⊙ λ = (µ ⊙ λ) ∨ (ν ⊙ λ).
(iv) Let x be an arbitrary element of G. If z 6= y, then χ(y, a)(z) = 0,
so we have
(χ(y, a) ⊙ λ)(x) = ∨z∈G(χ(y, a)(z) ∧ λ(z−1x))
= χ(y, a)(y) ∧ λ(y−1x)) = a ∧ λ(y−1x).
The proof of (v) is similar.
L. A. Kurdachenko, V. S. Yashchuk, I. Ya. Subbotin 131
(vi) If u ∈ G, b ∈ L, then (χ(y, a) ⊙ χ(u, b))(x) = a ∧ χ(u, b)(y−1x).
Recall that χ(u, b)(y−1x) = b if y−1x = u or x = yu and χ(u, b)(y−1x) = 0
if y−1x 6= u or x 6= yu. Thus
(χ(y, a) ⊙ χ(u, b))(x) =
{
a ∧ b, if x = yu
0, if x 6= yu.
Hence we obtain (vi).
(vi) Using the above arguments we obtain
(χ(x, a)⊙(γ ⊙ χ(x−1, a)))(y)
= ∨u,v,z∈G,uvz=yχ(x, a)(u) ∧ (γ(v) ∧ χ(x−1, a))(z)
= χ(x, a)(x) ∧ γ(x−1yx) ∧ χ(x−1, a)(x−1)
= a ∧ γ(x−1yx) ∧ a = a ∧ γ(x−1yx).
Let G be a group and γ ∈ LG. Then a surjective function γ is said to
be a group function on G if it satisfies the following conditions:
(GF 1) γ(xy) > γ(x) ∧ γ(y) for all x, y ∈ G,
(GF 2) γ(x−1) > γ(x) for every x ∈ G.
Let γ, κ group functions on G. If γ 6 κ, then we will say that γ is a
subgroup function of κ. This fact we will denote γ � κ.
Proposition 2. Let G be a group, L be a finite distributive lattice, γ ∈ LG,
and suppose that γ is a group function on G. Then the following assertions
hold:
(i) γ(x−1) = γ(x) for every x ∈ G (in order words, a function γ is
even).
(ii) γ(xy−1) > γ(x) ∧ γ(y) for all x, y ∈ G.
(iii) γ(xn) > γ(x) for every x ∈ G and every integer n.
(iv) γ(e) > γ(x) for every x ∈ G.
(v) Let λ, κ 6 γ, then λ ⊙ κ 6 γ, in particular, γ ⊙ γ 6 γ.
Proof. (i) We have x = (x−1)−1, so (GF 2) implies that γ(x) > γ(x−1),
which together with γ(x−1) > γ(x) gives γ(x) = γ(x−1) for every element
x ∈ G.
(ii) Let x, y be arbitrary elements of G. By (GF 1) γ(xy−1) > γ(x) ∧
γ(y−1), and by (i) γ(y−1) = γ(y), so that γ(xy−1) > γ(x) ∧ γ(y).
132 Lattice groups
(iii) Let x ∈ G. By (GF 1) γ(x2) = γ(xx) > γ(x) ∧γ(y) = γ(x). Using
ordinary induction, we obtain that γ(xn) > γ(x) for every n ∈ N. Suppose
now that n = −k where k ∈ N. Then xn = (x−1)k. By proved above
γ(xn) = γ((x−1)k) > γ(x−1) = γ(x).
(iv) Let x ∈ G. By (GF 1) we have
γ(e) = γ(xx−1) > γ(x) ∧ γ(x−1) = γ(x) ∧ γ(x) = γ(x).
(v) Let x be an arbitrary element of G. The inclusions λ, κ 6 γ imply
λ(y)∧κ(z) 6 γ(y)∧γ(z). Since γ is a group function, γ(y)∧γ(z) 6 γ(yz),
thus
(λ ⊙ κ)(x) = ∨y,z∈G,yz=x(γ(y) ∧ κ(z)) 6 ∨y,z∈G,yz=xγ(yz) = γ(x).
Proposition 3 (A criterion of group function). Let G be a group, L be
a finite distributive lattice and γ ∈ LG. Then γ is a group function on G
if and only if the following assertions hold:
(GF 3) χ(x, γ(x)) ⊙ χ(y, γ(y)) ⊆ γ for all x, y ∈ G.
(GF 4) χ(x−1, γ(x)) ⊆ γ for every x ∈ G.
Proof. Suppose first that γ is a group function. Clearly χ(x, γ(x)) ⊆ γ
and χ(y, γ(y)) ⊆ γ for all elements x, y ∈ G. Using Proposition 2 (v) we
obtain that
χ(x, γ(x)) ⊙ χ(y, γ(y)) ⊆ γ.
Let x be an arbitrary element of G. We have (χ(x−1, γ(x)))(x−1) =
γ(x). Since γ is a group function, γ(x) 6 γ(x−1). We note that if y 6= x−1,
then (χ(x, γ(x)))(y) = 0, so that (χ(x−1), γ(x)))(y) 6 γ(y) for every
y ∈ G. This means that χ(x−1, γ(x)) ⊆ γ.
Conversely, suppose that γ satisfies both conditions (GF 3) and (GF 4).
Let x, y be arbitrary elements of G. Then (GF 3) shows that χ(x, γ(x)) ⊙
χ(y, γ(y)) ⊆ γ. By Proposition 1 (vi),
(χ(x, γ(x)) ⊙ χ(y, γ(y)))(xy) = γ(x) ∧ γ(y).
The inclusion χ(x, γ(x)) ⊙ χ(y, γ(y)) ⊆ γ implies that (χ(x, γ(x)) ⊙
χ(y, γ(y)))(xy) 6 γ(xy), thus we obtain γ(x) ∧ γ(y) 6 γ(xy), and γ
satisfies (GF 1).
Let x ∈ G. Since χ(x−1, γ(x)) ⊆ γ, (χ(x−1, γ(x)))(y) 6 γ(y) for every
y ∈ G. In particular, (χ(x−1, γ(x)))(x−1) = γ(x) 6 γ(x−1), so that γ
satisfies (GF 2).
L. A. Kurdachenko, V. S. Yashchuk, I. Ya. Subbotin 133
Let G be a group and L be a finite distributive lattice. Consider
the Cartesian product A = G × L. Define the operation (multiplication)
on A by the following rule: (u, a)(v, b) = (uv, a ∧ b) for all u, v ∈ G,
a, b ∈ L. This operation is associative because multiplication in G and the
operation ∧ in L are associative. The pair (e,m) is the identity element
for this operation. The obtained above criterion allows us to transform
the definition of the group function in the following.
A nonempty subset Λ of G × L is called a lattice group over L if it
satisfies the following conditions:
(LG 1) if (x, a) ∈ Λ and b 6 a, then (x, b) ∈ Λ;
(LG 2) if (x, a), (y, b) ∈ Λ, then (x, a)(y, b) ∈ Λ;
(LG 3) if (x, a) ∈ Λ, then (x−1, a) ∈ Λ.
For every element x ∈ prG(Λ) put CΛ(x) = {a ∈ L|(x, a) ∈ Λ}.
Observe at once that a lattice group Λ defines a group function on
G. Indeed, for every element x ∈ prG(Λ) the set CΛ(x) is not empty. Put
λ(x) = ∨CΛ(x). If x /∈ prG(Λ), then put λ(x) = 0. Then λ is a function. If
u, v ∈ G and λ(u) = a, λ(v) = b, then (uv, a∧b) ∈ Λ by condition (LG 2).
It follows that λ(uv) > a ∧ b = λ(u) ∧ λ(v), so that λ satisfies (GF 1).
Similarly, let λ(u) = a, then (u−1, a) ∈ Λ by condition (LG 3). It follows
that λ(u−1) > a = λ(u), so that λ satisfies (GF 2).
Let Λ, Γ be the lattice groups over L. If Λ includes Γ, then we will
say that Γ is a lattice subgroup of Λ, and will denote this by Γ 6 Λ.
If γ is a defined by Γ group function, then γ � λ.
Clearly G × L is the greatest lattice group over L, and E = {(e, 0)}
is the least lattice group over L; the last lattice group is called trivial.
Furthermore, if a ∈ L, then {(e, b)|b 6 a} is a lattice group over L.
Every lattice group Λ includes prG(Λ) × {0}. For every subgroup H
of G the subset H × {0} is a lattice group. Recall that a subset M of L
is called a lower (respectively upper) segment of L, if from a ∈ M and
b 6 a (respectively a 6 b) it follows that b ∈ M.
If a ∈ L, then the subset {x|x ∈ L and x 6 a} (respectively {x|x ∈ L
and x > a}) is a lower segment (respectively upper segment) of L. It called
the principal lower (respectively upper) segment of L generated by a.
Consider some preliminary properties of the lattice groups.
Proposition 4. Let G be a group, L be a finite distributive lattice and S
be a family of lattice subgroups over L. Then intersection ∩S is a lattice
subgroup.
134 Lattice groups
Proof. The proof is almost obvious.
Proposition 5. Let G be a group, L be a finite distributive lattice and Λ
a lattice group. Then:
(i) prL(Λ) is a semigroup by operation ∧ with identity e(Λ) = ∨CΛ(1)
and zero 0. Moreover, prL(Λ) is the principal lower segment of L,
generated by e(Λ).
(ii) prG(Λ) is a subgroup of G. Conversely, for every subgroup H of
prG(Λ) the subset {(x, a)|(x, a) ∈ Λ and x ∈ H} = pr−1
G (H) is a
lattice subgroup of Λ.
(iii) If M is a lower segment of L, then {(x, a)|(x, a) ∈ Λ and a ∈ M}
is a lattice subgroup of Λ. In particular, pr−1
L (M) is a lattice group.
Proof. (i) Indeed, if a, b ∈ prL(Λ), then there are elements u, v ∈ G such
that (u, a), (v, b) ∈ Λ. Since Λ is a lattice group, (uv, a∧b) = (u, a)(v, b) ∈
Λ. It follows that a ∧ b ∈ prL(Λ). In particular, e(Λ) = ∨CΛ(e) ∈ prL(Λ).
Let a ∈ prL(Λ) and u be an element of G such that (u, a) ∈ Λ. Since
Λ is a lattice group, (u−1, a) ∈ Λ by condition (LG 3). Using (LG 2), we
obtain that (e, a) = (uu−1, a) = (uu−1, a ∧ a) = (u, a)(u−1, a) ∈ Λ. Hence
a ∈ C(e), which follows that a 6 e(Λ). In other words, e(Λ) is the greatest
element of prL(Λ).
Let c be an arbitrary element of L such that c6e(Λ). Since (e, e(Λ))∈Λ.
(e, c) ∈ Λ by condition (LG 1). It follows that prL(Λ) is the principal
lower segment of L, generated by e(Λ).
(ii) Let K = prG(Λ), u, v ∈ K. Then there are the elements a, b ∈ L
such that (u, a), (v, b) ∈ Λ. Since Λ is a lattice group, (uv, a ∧ b) =
(u, a)(v, b) ∈ Λ. It follows that uv ∈ K. If (u, a) ∈ Λ, then (u−1, a) ∈ Λ by
condition (LG 3), which follows that u−1 ∈ K. Hence K is a subgroup
of G.
Let now H be a subgroup of prG(Λ), (u, a), (v, b) ∈ pr−1
G (H). Since
Λ is a lattice group, (uv, a ∧ b) = (u, a)(v, b) ∈ Λ. The fact that H is a
subgroup implies that uv ∈ H, so that (uv, a ∧ b) ∈ pr−1
G (H). Since H is
a subgroup, then from u ∈ H it follows that u−1 ∈ H. Since Λ is a lattice
group, (u, a) ∈ Λ implies that (u−1, a) ∈ Λ. Hence (u−1, a) ∈ pr−1
G (H),
so that pr−1
G (H) satisfies the conditions (LG 2), (LG 3), and (uv, a∧ b) =
(u, a)(v, b) ∈ Λ. Hence K is a subgroup of G. Let (u, a) ∈ pr−1
G (H) and
b be an element of L such that b 6 a. Then (u, b) ∈ Λ and hence
(u, b) ∈ pr−1
G (H).
(iii) Let M is a lower segment of L, K a subgroup of G and M = K×M.
Then M is a lattice group. Indeed, if (x, a) ∈ M and b 6 a, then b ∈ M,
L. A. Kurdachenko, V. S. Yashchuk, I. Ya. Subbotin 135
because M is a lower segment of L. It follows that (x, b) ∈ M, so that M
satisfies (LG 1). Suppose that (x, a), (y, b) ∈ M. Since a ∧ b 6 b, a ∧ b ∈
M. The fact that K is a subgroup of G implies xy ∈ K, and hence
(xy, a ∧ b) ∈ M. We note that (xy, a ∧ b) = (x, a)(y, b), which shows that
M satisfies (LG 2). Finally, let (x, a) ∈ M. Since K is a subgroup of G,
x−1 ∈ K. Therefore (x−1, a) ∈ M, and M satisfies (LG 3).
Let again H = prG(Λ), then it is not hard to see that
{(x, a)|(x, a) ∈ Λ and a ∈ M} = H × M ∩ Λ.
Proposition 4 shows that this subset is a lattice subgroup of Λ.
Let Λ be a lattice group. Unlike abstract groups, a lattice group
can contains more than one idempotent. Moreover, Λ contains a pair
(1, a) for each element a ∈ prL(Λ). Indeed, let u be an element of G
such that (u, a) ∈ Λ. Since Λ is a lattice group, (u, a)(u−1, a) ∈ Λ. But
(u, a)(u−1, a) = (e, a ∧ a) = (e, a). It shows that a semigroup Λ can be a
group only in the case when prL(Λ) contains only one element a. Let b ∈ Λ
and b 6 a, then condition (LG 1) implies that (u, b) ∈ Λ. Hence a = b. In
other words, a is the least element of L, i.e. a = 0. Consequently, a lattice
group Λ is a group if and only if prL(Λ) = {0}. In this regard, we note
that the semigroup Λ may include many subsemigroups, which are groups
by multiplication. Indeed, let H be a subgroup of G and a ∈ L, then it
is not hard to see that the subset H × {a} is a group by multiplication.
Furthermore, for every a ∈ L the subset {(u, a)|(u, a) ∈ Λ} is also a group
by multiplication.
If Λ is a lattice subgroup over L, then put E(Λ) = {(e, b)|b 6 e(Λ)}.
Clearly E(Λ) is a lattice subgroup of Λ.
Let Γ be a lattice subgroup of Λ. The pair (e, e(Λ)) is an identity
element of Λ and (e, e(Γ)) is an identity element of Γ. Since Γ 6 Λ,
Proposition 5 shows that e(Γ) 6 e(Λ). We say that Γ is an unitary
lattice subgroup of Λ, if (e, e(Λ)) ∈ Γ. Every lattice subgroup of Λ can be
extended to an unitary lattice subgroup. Indeed, put Γu(Λ) = Γ∪{(e, b)|b 6
e(Λ)} = Γ ∪ E(Λ), then Γu(Λ) is a lattice group. In fact, if (u, a) ∈ Λ,
then (u, a)(e, b) = (u, a ∧ b). Since a ∧ b 6 a, (u, a ∧ b) ∈ Γ. It shows that
Γu(Λ) satisfies all conditions (LG 1)–(LG 3).
Let M be a subset of G × L and S be a family of all lattice groups,
including M. By Proposition 4, the intersection ∩S is a lattice group. It
called the lattice group generated by M and will be denoted by 〈M〉.
Let (x, a) ∈ G×L. If Λ is a lattice group containing (x, a), then it is not
hard to prove that (x, a)n = (xn, a∧ . . .∧a) = (xn, a) ∈ Λ for each positive
136 Lattice groups
integer n. By (LG 3), (x−1, a) ∈ Λ, and hence (e, a) = (x, a)(x−1, a) ∈ Λ.
From (x−1, a) ∈ Λ we obtain that (x, a)−n = (x−n, a) ∈ Λ, so that
{(xn, a)|n ∈ Z} ⊆ Λ. Let A be the principal lower segment of L, generated
by a. If b 6 a, then (LG 1) implies that (xn, b) ∈ Λ for each integer
n. Thus {(xn, b)|b 6 a, n ∈ Z} ⊆ Λ. It is not hard to check that the
subset {(xn, b)|b 6 a, n ∈ Z} is a lattice group. It follows that 〈(x, a)〉 =
{(xn, b)|b 6 a, n ∈ Z}.
Let Λ, Γ be the lattice subgroups. Define its product in the usual way:
put
ΛΓ = {(x, a)(y, b) = (xy, a ∧ b)|(x, a) ∈ Λ, (y, b) ∈ Γ}.
The following result is a rationale for this determination.
Proposition 6. Let G be a group, L be a finite distributive lattice and
γ, κ : G → L be functions. Then
γ ⊙ κ = ∪y∈Supp(γ),z∈Supp(κ)χ(y, γ(y)) ⊙ χ(z, κ(z)).
Proof. By definition we have
(γ ⊙ κ)(x) = ∨y,z∈G,yz=x(γ(y) ∧ κ(z)).
If y /∈ Supp(γ), then γ(y) = 0 and γ(y) ∧ κ(z) = 0. Similarly, if z /∈
Supp(κ), then κ(z) = 0, and again γ(y) ∧ κ(z) = 0. It follows that
(γ ⊙ κ)(x) = ∨y∈Supp(γ),z∈Supp(κ),yz=x(γ(y) ∧ κ(z)).
On the other hand, let ξ = ∪y∈Supp(γ),z∈Supp(κ)χ(y, γ(y)) ⊙ χ(z, κ(z)).
By Proposition 1, χ(y, γ(y)) ⊙ χ(z, κ(z)) = χ(yz, (γ(y) ∧ κ(z))). If x ∈
G and x = yz, then χ(yz, (γ(y) ∧ κ(z)))(x) = γ(y) ∧ κ(z), otherwise
χ(yz, (γ(y) ∧ κ(z)))(x) = 0. Therefore
ξ(x) = ∨y∈Supp(γ),z∈Supp(κ)(χ(yz, (γ(y) ∧ κ(z))))(x)
= ∨y∈Supp(γ),z∈Supp(κ),yz=x(γ(y) ∧ κ(z)) = (γ ⊙ κ)(x).
Since it is true for each x ∈ G,
γ ⊙ κ = ∪y∈Supp(γ),z∈Supp(κ)χ(y, γ(y)) ⊙ χ(z, κ(z)).
Corollary. Let G be a group, L be a finite distributive lattice, a ∈ L, and
κ : G → L be functions. Then for every x ∈ G
χ(x, a) ⊙ κ = ∪z∈Supp(κ)χ(x, a) ⊙ χ(z, κ(z)),
κ ⊙ χ(x, a) = ∪z∈Supp(κ)χ(z, κ(z)) ⊙ χ(x, a).
L. A. Kurdachenko, V. S. Yashchuk, I. Ya. Subbotin 137
Let λ : G → L be a function defined by Λ and γ : G → L be a function
defined by Γ. Consider a function κ : G → L defined by the product ΛΓ.
Let g be an arbitrary element of G. If g /∈ prG(ΛΓ), then κ(g) = 0. On
the other hand, let u, v be an arbitrary elements of G such that g = uv.
Since g /∈ prG(ΛΓ) = prG(Λ) prG(Γ), then either u /∈ prG(Λ), v /∈ prG(Γ),
or u ∈ prG(Λ) but v /∈ prG(Γ) or u /∈ prG(Λ) but v ∈ prG(Γ). In each of
these cases either λ(u) = 0 or γ(v) = 0, so that
∨u,v∈G,uv=g(λ(u) ∧ γ(v)) = 0 = κ(g).
Suppose now that g ∈ prG(ΛΓ), then κ(g) = ∨CΛΓ(g). Let again u, v be
arbitrary elements of G such that g = uv. If u /∈ prG(Λ) or v /∈ prG(Γ),
then (λ(u)∧γ(v)) = 0. Suppose that u ∈ prG(Λ) and v ∈ prG(Γ) and let a,
b be the elements of L such that (u, a), (v, b) ∈ L. We have (u, a)(v, b) =
(uv, a ∧ b). This shows that CΛΓ(g) = {a ∧ b|a ∈ CΛ(u), b ∈ CΓ(v)}. Since
λ(u) = ∨CΛ(u), γ(v) = ∨CΓ(v), CΛΓ(g) = λ(u) ∧ γ(v). In other words, in
this case we have also
κ(g) = ∨u,v∈G,uv=g(λ(u) ∧ γ(v)).
Thus κ = λ ⊙ γ. Thus, from the bulky and not very transparent product
of functions we come to the intuitively clear and convenient product of
subsets.
Let us now see how another important concept, the concept of normal
fuzzy subgroup can be transformed. Again, it should be recalled that we
use different terminology.
Let λ, κ : G → L be a group functions and κ � λ. We say that κ
is a normal subgroup function of λ, if κ(yxy−1) > κ(x) ∧ λ(y) for every
elements x, y ∈ G.
We will need the following criteria of normality.
Proposition 7. Let G be a group, L be a finite distributive lattice and
λ, κ : G → L be group functions such that κ � γ. Then the following
assertions are equivalent:
(i) κ is a normal subgroup function of γ;
(ii) χ(x, γ(x)) ⊙ κ ⊙ χ(x−1, γ(x)) � κ for every element x ∈ G;
(iii) χ(x, γ(x))⊙χ(y, κ(y))⊙χ(x−1, γ(x)) ⊆ κ for every elements x, y ∈G;
(iv) χ(x, a) ⊙ χ(y, b) ⊙ χ(x−1, a) ⊆ κ for every elements x, y ∈ G,
a 6 γ(x), b 6 κ(y).
138 Lattice groups
Proof. (i) ⇒ (ii). Suppose that κ is a normal subgroup function of λ.
For arbitrary element y ∈ G we consider the product χ(y, γ(y)) ⊙ κ ⊙
χ(y−1, γ(y)). Let x be an arbitrary element of G. From Proposition 1 we
obtain
(χ(y, γ(y)) ⊙ κ ⊙ χ(y−1, γ(y)))(x) = γ(y) ∧ κ(y−1xy).
Put u = y−1xy, then x = y(y−1xy)y−1 = yuy−1, so that
(χ(y, γ(y)) ⊙ κ ⊙ χ(y−1, γ(y)))(yuy−1) = γ(y) ∧ κ(u).
Since κ(u) ∧ γ(y) 6 κ(yuy−1), we obtain
(χ(y, γ(y)) ⊙ κ ⊙ χ(y−1, γ(y)))(yuy−1) 6 κ(yuy−1),
that is
(χ(y, γ(y)) ⊙ κ ⊙ χ(y−1, γ(y)))(x) 6 κ(x).
Since this is valid for every element x ∈ G,
χ(y, γ(y)) ⊙ κ ⊙ χ(y−1, γ(y)) � κ.
(ii) ⇒ (iii). Indeed, Corollary to Proposition 6 shows that
χ(y, γ(y)) ⊙ κ ⊙ χ(y−1, γ(y))
= ∪z∈Supp(κ)χ(y, γ(y)) ⊙ χ(z, κ(z)) ⊙ χ(y−1, γ(y)).
Hence the inclusion χ(y, γ(y)) ⊙ κ ⊙ χ(y−1, γ(y)) � κ implies that
χ(y, γ(y)) ⊙ χ(z, κ(z)) ⊙ χ(y−1, γ(y)) ⊆ κ for every elements y, z ∈ G.
(iii) ⇒ (iv). Indeed, Proposition 1 shows that
χ(x, γ(x)) ⊙ χ(y, κ(y)) ⊙ χ(x−1, γ(x)) = χ(xyx−1, γ(x) ∧ κ(y)).
We have
χ(x, a) ⊙ χ(y, b) ⊙ χ(x−1, a) = χ(xyx−1, a ∧ b) ⊆ χ(xyx−1, γ(x) ∧ κ(y)).
(iv) ⇒ (i). Using again Proposition 1, we obtain that
χ(x, γ(x)) ⊙ χ(y, κ(y)) ⊙ χ(x−1, γ(x)) = χ(xyx−1, γ(x) ∧ κ(y)).
Now (vi) shows that χ(xyx−1, γ(x) ∧ κ(y)) ⊆ κ. Then
γ(x) ∧ κ(y) = χ(xyx−1, γ(x) ∧ κ(y))(xyx−1) 6 κ(xyx−1).
This means that κ is a normal subgroup function of γ.
L. A. Kurdachenko, V. S. Yashchuk, I. Ya. Subbotin 139
Proposition 7 leads us to the following analogue of normality in lattice
groups.
Let Γ be a lattice subgroup of Λ. We say that Γ is a normal lattice
subgroup of Λ, if (y−1, b)(x, a)(y, b) ∈ Γ for all pairs (y, b) ∈ Λ, (x, a) ∈ Γ.
We remark that (y−1, b)(x, a)(y, b) = (y−1xy, a∧b). At once this shows
that if Γ a normal lattice subgroup of Λ, then prG(Γ) is a normal subgroup
of prG(Λ). Conversely, suppose that H is a normal subgroup of G and
ΛH = {(x, a) ∈ Λ|x ∈ H}. Then (y−1, b)(x, a)(y, b) = (y−1xy, a ∧ b) ∈ Λ
for each pair (y, b) ∈ Λ. Since H is normal in G, y−1xy ∈ H, so that
(y−1, b)(x, a)(y, b) ∈ ΛH .
Let M be a lower segment of L . Then Proposition 5 proves that
Λ[M] = {(x, a)|(x, a) ∈ Λ and a ∈ M} is a lattice subgroup of Λ. Λ[M] is
called an M-layer of Λ. We note that Λ[M] is a normal lattice subgroup
of Λ. In fact, let (x, a) ∈ Λ[M] and (y, b) ∈ Λ, then (y−1, b)(x, a)(y, b) =
(y−1xy, a ∧ b). Since a ∧ b 6 a, a ∈ M and M is a lower segment of L,
a ∧ b ∈ M. Thus (y−1, b)(x, a)(y, b) ∈ Λ[M].
If Γ is a normal lattice subgroup of Λ, then Γu(Λ) is a normal lattice
subgroup of Λ. Indeed, let (x, a) ∈ Γu(Λ) and (y, b) ∈ Λ. If x 6= e, then
(x, a) ∈ Γ and (y−1, b)(x, a)(y, b) ∈ Γ. If x = e, then (y−1, b)(e, a)(y, b) =
(1, a ∧ b) ∈ E(Λ).
The layers of lattice group play a very important role. Especially it is
useful in the case when prL(Λ) is a chain. This case arises in theory of
fuzzy group when a group G is finite. Suppose that | prL(Λ)| = k. Then
prL(Λ) is isomorphic (as an ordered set) to the set Ch[1, k] = {1, 2, . . . , k}
with the natural ordering 1 6 2 6 . . . 6 k. In this case, we will say that
Λ is a lattice group over Ch[1, k].
For this case we construct some natural series of subgroups both in
the lattice group Λ and in prG(Λ). The subset {1} is the lower segment
of Ch[1, k], and therefore the {1}-layer Λ[1] of Λ is a lattice subgroup
of Λ. If (u, m) ∈ Λ, then (u, 1) ∈ Λ by condition (LG 1). This im-
plies that prG(Λ) = prG(Λ[1]). For every m, 1 6 m 6 k, the subset
Km = {(u, m)|(u, m) ∈ Λ} is the subgroup by multiplication, so that
H(m) = prG(Km) is a subgroup of H(1) = prG(Λ). A subgroup H(m) is
called the m-hoop of Λ. From (u, m) ∈ Λ we obtain (u, m − 1) ∈ Λ by
condition (LG 1). This implies the inclusion H(m) 6 H(m − 1), so we
obtain the following descending series of subgroups
H(1) > H(2) > . . . > H(k).
Clearly the mapping u → (u, m), u ∈ H(m), is an isomorphism of H(m)
on Km for each m, 1 6 m 6 k.
140 Lattice groups
Figuratively speaking, the pictured structure of a lattice group over
Ch[1, k] reminds the cake “Napoleon”. Here the groups play the role of
the cakes lays, and the idempotents play the role of cream lays. Indeed,
in the first step, by above remarked the Λ[1] is a normal lattice subgroup
of Λ. We have seen also that Λ[1] is a group by multiplication (moreover,
it is isomorphic to prG(Λ)). Now add the cream: put Λ1 = Λ[1] ∪ {(e, 2)}.
It is not hard to see, that Λ1 is a normal lattice subgroup of Λ. Next step:
consider the {1, 2}-layer Λ[1, 2] of Λ, which is a normal lattice subgroup
of Λ. We note that Λ1 6 Λ[1, 2], moreover Λ1 is a normal lattice subgroup
of Λ. For every element (x, j) ∈ Λ[1, 2] denote by (x, j)Λ1 the product
{(x, j)}Λ1. This subset is called a coset by Λ1. Since (x, j) ∈ Λ[1, 2],
j 6 2, so that (x, j) = (xe, j ∧ 2) = (x, j)(e, 2) ∈ (x, j)Λ1. It follows that
Λ[1, 2] is an union of all subsets (x, j)Λ1. Suppose that (x, j)Λ1 6= Λ1.
Then x 6= e and j = 2. Thus we can see that the equality (x, 2) =
(y, 2)(z, m) where (z, m) ∈ Λ1 is possible only in the case when m = 2.
In turn, the single pair of Λ1, whose second component is equal to 2,
is the pair (e, 2). Hence (x, 2) = (y, 2)(e, 2), so that x = y. In other
words, the equality (x, 2)Λ1 = (y, 2)Λ1 is possible only in the case, when
x = y. Consider the product of subsets ((x, 2)Λ1)((y, 2)Λ1). Its arbitrary
element has a form (x, 2)(u, j)(y, 2)(v, m) where (u, j), (v, m) ∈ Λ1. If
j = 1 or m = 1, then (x, 2)(u, j)(y, 2)(v, m) = (xuyv, 1) ∈ Λ1. Hence if
(x, 2)(u, j)(y, 2)(v, m) /∈ Λ1, then j = m = 2. But it is possible only if
u = v = e. In this case, (x, 2)(u, j)(y, 2)(v, m) = (xy, 2). In turn it follows
that ((x, 2)Λ1)((y, 2)Λ1) = (xy, 2)Λ1. Hence the set of all cosets by Λ1
becomes a semigroup. Moreover, this semigroup is a group, because it
has an identity element (e, 2)Λ1 = Λ1, and for every coset (x, 2)Λ1 we
have (x−1, 2)Λ1(x, 2)Λ1 = (e, 2)Λ1 = (x, 2)Λ1(x−1, 2). Therefore we can
talk here about a factor-group of a lattice group Λ[1, 2] by the normal
lattice subgroup Λ1. For it we will use a common notation Λ[1, 2]/Λ1.
We emphasize that here we are talking about a factor-group, rather
than a lattice factor-group. It is our special selection provides such an
opportunity; in general, is not always the family of cosets by normal
lattice subgroup is a group or a lattice group.The mapping Φ, defined
by the rule Φ((x, 2)) = (x, 2)Λ1, (x, 2) ∈ K2, is an epimorphism. As we
have seen early, the equality (x, 2)Λ1 = Λ1 is possible only in the case
when x = e, which shows that Φ is an isomorphism. Since K2
∼= H(2), we
obtain that Λ[1, 2]/Λ1 is isomorphic to the 2-hoop of Λ.
Adding the next lay of the cream {(e, 3)} to Λ[1, 2], we come to the
normal lattice subgroup Λ2 = Λ[1, 2] ∪ {(e, 3)}, and then we cover it with
the next lay of cake, i.e. extend Λ2 to the {1, 2, 3}-layer Λ[1, 2, 3] of Λ,
L. A. Kurdachenko, V. S. Yashchuk, I. Ya. Subbotin 141
which is a normal lattice subgroup of Λ. Using the above arguments, we
shows that a family of cosets (x, 3)Λ2 is a group by multiplication and
this group is isomorphic to the 2-hoop of Λ. And so on. As the result we
obtain the sequences
Λ0 = {(e, 1)} 6 Λ[1] 6 Λ1 6 Λ[1, 2] 6 Λ2 6 Λ[1, 2] 6 . . . 6 Λk−1 6 Λ
of normal lattice subgroups such that Λm = Λ[1, . . . , m] ∪ {(e, m + 1)},
and Λ[1, . . . , m + 1]/Λm
∼= H(m + 1), 0 6 m 6 k − 1.
Note, that in the theory of fuzzy groups we could not find any similar
description of a general structure of a fuzzy group γ for the case when
Im(γ) is finite.
References
[1] Goguen J.A., L-Fuzzy Sets, Journal of Math. Analysis and Applications., N.18,1967,
pp.145-174.
[2] Kurdachenko L.A., Grin K.O., Turbay N.A., On normalizers in fuzzy groups, Algebra
and Discrete Mathematics., N.15,2013, pp.23-36.
[3] Kurdachenko L.A., Otal J., Subbotin I.Ya., On permutable fuzzy subgroups, Serdica
Mathematical Journal., N.39,2013, pp.83-102.
[4] Mordeson J.N., Nair P.S., Fuzzy Mathematics, Springer: Berlin., 2001.
[5] Mordeson J.N., Bhutani K.R., Rosenfeld A., Fuzzy Group Theory, Springer: Berlin,
2005.
[6] Zadeh L.A., Fuzzy sets, Information Control., N.8, 1965, pp.338-353.
Contact information
Leonid A. Kurdachenko,
Viktoriia S. Yashchuk
Department of Algebra, Oles Honchar
Dnipropetrovsk National University, 72
Gagarin Av., Dnepropetrovsk, Ukraine
49010
E-Mail(s): lkurdachenko@i.ua,
ViktoriiaYashchuk@mail.ua
Igor Ya. Subbotin Department of Mathematics and Natural
Sciences, National University, 5245 Pacific
Concourse Drive, LA, CA 90045, USA
E-Mail(s): isubboti@nu.edu
Received by the editors: 02.04.2015
and in final form 02.04.2015.
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