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CATEGORICAL PROPERTIES OF SEQUENTIALLY DENSE MONOMORPHISMS OF SEMIGROUP ACTS

TAIWANESE JOURNAL OF MATHEMATICS (Impact Factor: 0.67). 05/1388; 15(2):22-23.

ABSTRACT Let M be a class of (mono)morphisms in a category A. To study mathematical notions, such as injectivity, tensor products, flatness, one needs to have some categorical and algebraic information about the pair (A,M). In this paper we take A to be the category Act-S of S-acts, for a semigroup S, and M d to be the class of sequentially dense monomor-phisms (of interests to computer scientists, too) and study the categori-cal properties, such as limits and colimits, of the pair (A,M). Injectivity with respect to this class of monomorphisms have been studied by Giuli, Ebrahimi, and the authors and got information about injectivity relative to monomorphisms.

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    ABSTRACT: To study mathematical notions, such as injectivity with respect to the class M of (mono)morphisms in a category A, one needs to have some categorical and algebraic information about the pair (A, M). In this paper, we take A to be the category Act-S of acts over a semigroup S, C to be an arbitrary closure operator in the category Act-S, and M d to be the class of C-dense monomorphisms resulting from a closure operator C and first study some categorical properties of the pair (Act-S, M d). Then injectivity with respect to the class of C-dense monomorphisms is studied. The class of sequentially dense monomorphisms resulting from a special closure operator (sequential closure operator) and injectivity with respect to this class of monomorphisms have been studied by Giuli, Ebrahimi, Mahmoudi and the author. Some of these results generalize some of the results about the class of sequentially dense monomorphisms.
    Asian-European Journal of Mathematics 09/2013; 3(5).
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    ABSTRACT: An important notion related to injectivity with respect to monomorphisms or any other class M of morphisms in a category A is essentialness. In this paper, taking A to be the category of right acts over a semigroup S, C to be an arbitrary clo-sure operator in the category Act-S, and M d to be the class of C-dense monomorphisms resulting from a closure operator C, we study the properties of M d -essential monomor-phisms and we show the existence of a maximal M d -essential extension for any given act. Finally, the behavior of M d -injectivity in the sense that the three so called Well-behavedness propositions hold is studied. We show that the idempotency and weak hereditariness of a closure operator C are sufficient, but not necessary, conditions for the well-behavedness of M d -injectivity. The class of sequentially dense monomorphisms resulting from a special closure operator (sequential closure operator) and injectivity with respect to this class of monomorphisms have been studied by Giuli, Ebrahimi, Mahmoudi, Moghaddasi, and the author. Some of these results generalize some of the results about the class of sequentially dense monomorphisms.
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    ABSTRACT: s-dense monomorphisms and injectivity with respect to these monomorphisms were rst introduced and studied by Giuli for acts over the monoid (N ∞ , min). Ebrahimi, Mahmoudi, Moghaddasi, and Shahbaz generalized these notions to acts over a general semigroup. In this paper, we study atness with respect to the class of s-dense monomorphisms. The theory of atness properties of acts over monoids has been of major interest over the past some decades, but so far there are not any papers published on this subject that relate specically to the class of s-dense monomorphisms. We give some sucient conditions for s-dense atness of semigroup acts. Also, we characterize a large number of semigroups over which s-dense atness coincides with atness. This gives a useful criterion for atness of acts over such semigroups. In fact it is shown that the study of s-dense atness is also useful in the study of ordinary atness of acts.
    Quasigroups and Related Systems. 10/2013; 21:201-206.

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