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On Isomorphism Theorems for MI-Groups 8th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2013) On Isomorphism Theorems for MI-groups Michal Holčapek, Michaela Wrublová, Martin Štěpnička Centre of Excellence IT4Innovations, division of University of Ostrava Institute for Research and Applications of Fuzzy Modeling 30. dubna 22, 701 03 Ostrava 1, Czech Republic {michal.holcapek, michaela.wrublova, martin.stepnicka}@osu.cz Abstract This led the authors in [3, 4] to introduced MI- algebras (e.g. MI-monoids, MI-groups, MI-rings or The theory of MI-algebras (“Many Identities”- MI-fields) that naturally generalize standard alge- algebras) has been introduced by M. Holčapek and braic structures (monoids, groups, rings or fields) in M. Štěpnička recently. These structures motivated such a way that they employ so called set of pseu- by an algebraic formalizations of distinct arith- doidentities, i.e., a set of such elements that preserve metics of fuzzy numbers, generalize the standard some but not all properties of classical identities. structures (monoids, groups, fields etc.) by em- It should be noted that the arithmetic of inter- ploying a whole set of identity-like elements called vals as special fuzzy sets or extended intervals in- pseudoidentitites. This leads to natural questions: volving infinite intervals has been intensively inves- which properties of classical algebraic structures re- tigated by Markov [7]. The interesting arithmetic lated to arithmetics of crisp numbers are preserved of stochastic intervals can be found in [8]. also in the case of MI-structures. This paper fo- In this paper, we focus only of MI-groups as cuses on MI-groups, particularly on the properties a generalization of one of the most natural alge- of their homomorphisms. We show that three cru- braic structures. We investigate whether the same cial theorems of isomorphism that are valid in the or similar concepts, theorems and properties, that classical case, are under some conditions also valid are known for groups, hold also in the case of MI- in the theory of MI-groups. groups. The structure of the paper is as follows. Assuming that readers are familiar with the clas- Keywords: many identities groups, algebraic struc- sical theory of groups we only refer to a relevant tures, fuzzy numbers, isomorphism theorems source of detailed information [6] and for the sake of brevity, omit any introduction to this theory. We 1. Introduction recall the basic definitions, propositions and theo- rems related to MI-groups. Then we describe the The concept of MI-groups was introduced by concept of homomorphism of MI-groups, the intro- M. Holčapek and M. Štěpnička in [3, 4] and later duce full MI-subgroups, normal full MI-subgroups on elaborated in [5]. The motivation for the in- and quotient MI-groups. Finally, we state condi- troduction of such algebraic structures came from tions under which the three isomorphism theorems the arithmetics of fuzzy numbers. It is well-known are valid. that standard arithmetics of fuzzy numbers that are based on the Zadeh’s extensional principle do lack some classical properties of the arithmetic of crisp 2. MI-groups numbers. Particularly, the following equalities MI-group is an algebraic structure based on a gen- (i) x + (−x) = 0, eralization of concept of monoid satisfying the can- (ii) xx−1 = 1, cellation law which is endowed with a monoidal iso- morphism representing the inversion. As we have are not generally satisfied, for x being a fuzzy num- mentioned above the main idea of our generaliza- ber and 0, 1 as the crisp zero and unit element, tion of groups consists in the introduction of a set respectively. In other words, the problem lies in of pseudoidentities which possess similar properties the non-existence of inverse elements for both arith- like the identity (neutral) element. As a conse- metic operations on fuzzy numbers. But obviously, quence of the definition we obtain that groups as this is just the consequence of the crispness of both well as commutative monoids equipped with an in- neutral (identity) elements. However, we also know volutive operation representing inversion proposed that in the arithmetics of fuzzy numbers: by Bica in [1] (see [2]) form special subclasses of x + (−x) = 0˜, xx−1 = 1˜, MI-groups. In what follows, we briefly recall basic definitions where 0˜, 1˜ are fuzzy numbers that are located close and properties used in this paper. For details in- to 0 and 1 and also from the algebraic point of view, cluding an extensive motivation and examples, we they work similarly to 0 and 1, respectively. refer to [3, 4, 5]. © 2013. The authors - Published by Atlantis Press 764 2.1. Basic definitions Definition 4. Let G = (G, ◦,E) be an MI-monoid. A dual MI-monoid to G is Gop = (G, ◦op,E), where Definition 1. A triplet (G, ◦,E) is said to be an ◦op is given by x ◦op y = y ◦ x. MI-monoid if E is a non-empty subset of G and ◦ is a binary operation on G such that for any x, y, z ∈ Definition 5. An MI-monoidal isomorphism f of G and a, b ∈ E the following conditions are satisfied: G onto Gop that satisfies (M1) x ◦ (y ◦ z) = (x ◦ y) ◦ z, (G1) f(x) ◦ x ∈ E, (M2) ∃e ∈ E, ∀x ∈ G : x ◦ e = e ◦ x = x, (G2) f(x) ◦ x = x ◦ f(x), (M3) a ◦ b ∈ E, (M4) x ◦ a = a ◦ x. (G3) f(f(x)) = x Elements from E are called pseudoidentity elements for any x ∈ G is called an f-inversion (inversion, (pseudoidentities, for short) and a pseudoidentity e for short) in G. satisfying (M2) is called a (strong) identity element. An MI-monoid G is said to be abelian or commu- For the sake of simplicity and conventional man- tative if x ◦ y = y ◦ x holds for any x, y ∈ G. ner, the inversion will be denoted using f(x) = x−1. An element x ∈ G with x−1 = x is said to be sym- One can see that each MI-monoid (G, ◦,E) is a metric in G. In the group theory, the symmetric monoid (G, ◦) with an abelian submonoid E (char- elements are called the elements of the second or- acterizing pseudoidentity elements) which elements der, i.e., it holds xx = e. commute with each element of G. Each monoid (G, ◦) is an MI-monoid (G, ◦, {e}), where e is the Theorem 1. Let PG be the least submonoid of G identity, therefore, MI-monoids generalize monoids. −1 containing the set {xx | x ∈ G}. Then PG is The concept of MI-monoidal homomorphism is an abelian submonoid of symmetric elements of the defined in a common manner. monoid (G, ◦). Definition 2. Let G and H be MI-monoids. A −1 mapping f : G → H is a homomorphism of MI- Sketch of the proof: Put X = {xx | x ∈ G} monoids provided and define PG = {a1a2 ··· an | ai ∈ X, n ∈ N}, where N denotes the set of all natural numbers. It (HM1) f(xy) = f(x)f(y) for all x, y ∈ G, is easy to verify that PG is the least submonoid of G containing X. 2 (HM2) f(eG) = eH , The set P could be interpreted as the least ad- (HM3) f(x) ∈ E for all x ∈ E , G H G missible set of pseudoidentities containing “active where eG and eH (EG and EH ) denote the strong pseudoidentities”, it means that they are the prod- identity (the set of strong identity and pseudoiden- uct of an element and its inversion in G. Now, we tities elements) of G and H, respectively. If f is can proceed to introduce the concept of MI-groups. injective, f is said to be a monomorphism. If f is surjective, f is said to be a epimorphism. If f is Definition 6. An MI-monoid endowed with inver- bijective and f(EG) = {f(a) | a ∈ EG} = EH , f sion satisfying the cancellation law is called an MI- is said to be an isomorphism. In this case G and group. H are said to be isomorphic (written G =∼ H). A homomorphism f : G → G is called an endomor- It should be noted that the algebraic structure for phism and an isomorphism f : G → G is called an fuzzy numbers defined by Bica in [1] is an abelian automorphism. MI-group with E defined as the set of all symmet- ric elemets of G. Note that not all symmetric el- Definition 3. A MI-monoid G satisfies the cancel- ements in G are contained in PG. It means that lation law if Bica’s definition is a special example of MI-groups. xy = zy implies x = z (1) Another example shows a relationship between MI- groups and fuzzy groups assuming that the lattice holds for any x, y, z ∈ G operations satisfy the cancellation law. In contrast to the group structure, where the in- Example 1. Let us recall that a ?-fuzzy subgroup verse elements are introduced internally, our defi- of a group (G, ◦, e) is a fuzzy set H : G → [0, 1] nition needs an external unary operation defining satisfying the following axioms ([9]): the inverse elements. In [2], the inverse elements are defined using an involutive monoidal automor- (i) H(x) ?H(y) ≤ H(x ◦ y) for any x, y ∈ G, phism. Since we deal here with MI-monoids being not abelian in general, we consider an MI-monoidal (ii) H(x) ≤ H(x−1) for any x ∈ G, isomorphism of MI-monoid onto its dual which only reverses the order of operands. (iii) H(e) = 1, 765 where ? is usually the minimum operation, but also Sketch of the proof: It is easy to verify that eG ∈ −1 left-continuous t-norms are permitted.
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