### $\omega$-semigroups

C.H. Applebaum
<span title="">1974</span> <i title="Rocky Mountain Mathematics Consortium"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/77ppzerlsvfczgvezw53uqeqni" style="color: black;">Rocky Mountain Journal of Mathematics</a> </i> &nbsp;
§1. Introduction. Let € stand for the set of non-negative integers (numbers), V for the class of all subcollections of € (sets), A for the set of isols, and il for the class of all recursive equivalence types (RET). The relation of inclusion is denoted C, a recursively equivalent to ß by a -ß, for sets a and ß, and the RET of a by Req (a). For the purpose of this paper we say a semigroup is an ordered pair (a, p), where (i) a C e and (ii) p is a semigroup operation (i.e., an associative binary
more &raquo; ... ultiplication) on a X a. An co-semigroup is a semigroup (a, p), where p can be extended to a partial recursive function of two variables. The concept of an co-semigroup is a recursive analogue of a semigroup and is a generalization of an co-group. In this paper, the author shows (T2), that there are co-semigroups which are groups but not co-groups; but that all periodic co-semigroups which are groups, are co-groups (Tl). Theorems T3, T5, T6, T7, and Til give conditions for an co-semigroup to be an co-group. The recursive analogues of regular semigroup, inverse semigroup, and right group [ co-regular cosemigroup, inverse co-semigroup, and co-right group] are studied in sections §5, §6, §8 respectively, with particular attention paid to T(a), the analogue of the regular semigroup of all mappings from a into a, and 1(a), the analogue of the symmetric inverse semigroup on a. Theorems T17 and T22 relate co-regular co-semigroups to co-groups and T28 relates co-regular co-semigroups to inverse co-semigroups. In T42 and T43 we have two nice characterizations of an co-right group and T45 shows that a periodic co-semigroup that is a right group is an coright group. Finally section §7 gives a brief introduction to the cohomomorphism theory of co-semigroups. The author wishes to thank the referee for his helpful suggestions. §2. Basic concepts and notations. The reader of this paper is assumed to be familiar with the notation and basic results of [2], [6], and [7].
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