Yazar "Datuashvili, Tamar" seçeneğine göre listele

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    Actions and semi-direct products in categories of groups with action
    (Hacettepe University, 2023) Datuashvili, Tamar; Şahan, Tunçar
    Derived actions in the category of groups with action on itself Gr• are defined and de-scribed. This category plays a crucial role in the solution of two problems of Loday stated in the literature. A full subcategory of reduced groups with action rGr• of Gr• is intro-duced, which is not a category of interest but has some properties, which can be applied in the investigation of action representability in this category; these properties are similar to those, which were used in the construction of universal strict general actors in the category of interest. Semi-direct product constructions are given in Gr• and rGr• and it is proved that an action is a derived action in Gr• (resp. rGr•) if and only if the corresponding semi-direct product is an object of Gr• (resp. rGr•). The results obtained in this paper will be applied in the forthcoming paper on the representability of actions in the category rGr•.
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    From cssc-crossed modules to categorical groups
    (Tbilisi Centre for Mathematical Sciences, 2024) Datuashvili, Tamar; Mucuk, Osman; Alemdar, Nazmiye; Şahan, Tunçar
    For any cssc-crossed module a category is constructed, equipped with a structure and proved that this is a coherent categorical group. Together with a result of the previous paper, where to any categorical group the cssc-crossed module is associated, this construction will enable us to prove an equivalence between the categories of categorical groups and of cssc-crossed modules in the sequel to this paper.
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    Groups up to congruence relation and from categorical groups to c-crossed modules
    (Springer Science and Business Media Deutschland GmbH, 2020) Datuashvili, Tamar; Mucuk, Osman; Şahan, Tunçar
    We introduce a notion of c-group, which is a group up to congruence relation and consider the corresponding category. Extensions, actions and crossed modules (c-crossed modules) are defined in this category and the semi-direct product is constructed. We prove that each categorical group gives rise to a c-group and to a c-crossed module, which is a connected, special and strict c-crossed module in the sense defined by us. The results obtained here will be applied in the proof of an equivalence of the categories of categorical groups and connected, special and strict c-crossed modules.
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    Pentactions and action representability in the category of reduced groups with action
    (De Gruyter Open Ltd, 2023) Datuashvili, Tamar; Şahan, Tunçar
    A notion of pentaction of any object in the category rGr .{\mathbf{rGr}^{\bullet}} of reduced groups with action is introduced. The operations are defined in the set »e - - ac - (A) {\mathsf{Pentact}(A)} of pentactions of an object A of rGr.{\mathbf{rGr}^{\bullet}}. It is proved that if an object A is perfect with zero weak stabilizer in the sense defined in this paper, then »e-ac-(A) {\mathsf{Pentact}(A)} is an object of rGr. {\mathbf{rGr}^{\bullet}}, it has a derived action on A, the object A is action representable and »e - - ac -(A) {\mathsf{Pentact}(A)} represents all actions on A.