The power graph of a group [Formula: see text] is the graph whose vertex set is [Formula: see text] and two distinct vertices are adjacent if one is a power of the other. This paper investigates the minimal separating sets of power graphs of finite groups. For power graphs of finite cyclic groups, certain minimal separating sets are obtained. Consequently, a sharp upper bound for their connectivity is supplied. Further, the components of proper power graphs of [Formula: see text]-groups are studied. In particular, the number of components of that of abelian [Formula: see text]-groups are determined.

Abstract. This work investigates the rank properties of A + (Bn), the additive semigroup reduct of affine near-semiring over Brandt semigroup Bn. In this connection, this work reports the ranks r 1 , r 2 , r 3 and r 5 of A + (Bn) and identifies a lower bound for the upper rank r 4 (A + (Bn)). While this lower bound is found to be the r 4 (A + (Bn)) for n ≥ 6, in other cases where 2 ≤ n ≤ 5, the upper rank of A + (Bn) is still open for investigation.

In order to study the axiomatization of the if-then-else construct over possibly non-halting programs and tests, the notion of C-sets was introduced in the literature by considering the tests from an abstract C-algebra. This paper extends the notion of C-sets to C-monoids which include the composition of programs as well as composition of programs with tests. For the class of C-monoids where the C-algebras are adas a canonical representation in terms of functional C-monoids is obtained.2010 Mathematics Subject Classification. 08A70, 03G25 and 68N15.There are multiple studies (e.g., see [3,7,13,14]) on extending two-valued Boolean logic to three-valued logic. However McCarthy's logic (cf. [17]) is distinct in that it models the short-circuit evaluation exhibited by programming languages that evaluate expressions in sequential order, from left to right. In [6] Guzmán and Squier gave a complete axiomatization of McCarthy's three-valued logic and called the corresponding algebra a C-algebra, or the algebra of conditional logic. While studying if-then-else algebras in [15], Manes defined an ada (Algebra of Disjoint Alternatives) which is essentially a C-algebra equipped with an oracle for the halting problem.Jackson and Stokes in [11] studied the algebraic theory of computable functions, which can be viewed as possibly non-halting programs, together with composition, if-then-else and while-do. In this work they assumed that the tests form a Boolean algebra. Further, they demonstrated how an algebra of non-halting tests could be constructed from Boolean tests in their setting. Jackson and Stokes proposed an alternative approach by considering an abstract collection of non-halting tests and posed the following problem:Characterize the algebras of computable functions associated with an abstract C-algebra of non-halting tests.The authors in [19] have approached the problem by adopting the approach of Jackson and Stokes in [10]. The notion of a C-set was introduced through which a complete axiomatization for if-then-else over a class of possibly non-halting programs and tests, where tests are drawn from an ada, was provided.In this paper, following the approach of Jackson and Stokes in [10], we extend the notion of C-sets to include composition of possibly non-halting programs and of these programs with possibly non-halting tests. This object is termed a Cmonoid and we show that every C-monoid where the tests are drawn from an ada is embeddable in a canonical model of C-monoids, viz., functional C-monoids.

Using holonomy decomposition, the absence of certain types of cycles in automata has been characterized. In the direction of studying the structure of automata with cycles, this paper focuses on a special class of semi-flower automata and establish the holonomy decomposition of certain circular semiflower automata. In particular, we show that the transformation monoid of a circular semi-flower automaton with at most two bpis divides a wreath product of cyclic transformation groups with adjoined constant functions.

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