Many identities group (MI-group, for short) is an algebraic structure
generalizing the group structure where an involutive anti-automorphism
satisfying certain properties is used instead of the standard group
inversion. The concept of MI-group has been introduced by Holčapek and
Stěpnička in the article [1], which is devoted to algebraic structures
that unify the fundamental properties of arithmetics with vaguely
specified quantities (e.g., stochastic or fuzzy quantities). In [2] and
later in [3], we proposed the concept of normal MI-subgroups assuming
that the MI-subgroups contain the set of all pseudo identity elements of
the given MI-group (consider the elements of type xx^{-1} for x that
belongs to G, where ^{-1} is an involute anti-automorphism on a monoid
G). These MI-subgroups are called full. A consequence of our restriction
to full normal MI-subgroups is the fact that the quotient MI-groups
induced by this type of normal MI-subgroups have a group structure, i.e.,
they are isomorphic to groups. In [3] (see also [2]), we also proved a
version of three isomorphism theorems for MI-groups under the mentioned
restriction. Then, a natural question appeared whether the same or
similar results could be done also for non-full MI-subgroups. In the
presentation, we provide a answer to this question.
[1] M. Holčapek and M. Štěpnička. MI-algebras: a new framework for
arithmetics of (extensional) fuzzy numbers. Fuzzy Sets and Systems 257
(2014) 102-131.
[2] M. Holčapek, M. Wrublová, and M. Štěpnička. On isomorphism theorems
for MI-groups. Proceedings of the 8th conference of the European Society
for Fuzzy Logic and Technology (EUSFLAT), Atlantis Press (2013) 764-771.
[3] M. Holčapek, M Wrublová, and M. Bacovský. Quotient MI-groups. Fuzzy
sets and systems 283 (2016) 1-25.