We show that the span of an arbitrary simple closed curve X does not exceed the span of any starlike curve contained in the closure of the unbounded component of the complement of X.

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For a subsemigroup T of a semigroup S, Reg(T) denotes the set of regular elements of T, LReg(T) the set of left regular elements of T and reg(T) the set of elements of T which are regular in S. Characterizations of a semigroup S for which reg(Se) = Reg(Se) for each idempotent element e of S have been given in [3]. This type of semigroups is the semigroups S in which each element of the subsemigroup Se of S which is regular in S is a left regular element of Se for every idempotent element e of S. Moreover, this type of semigroups is the semigroups S in which the regular elements are left regular, equivalently the sets of regular and completely regular elements coincide [3]. In the present paper we prove that the type of semigroups mentioned above is actually the semigroups in which reg(Sa) = reg(Sa) for every a ∈ S.

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The upwards-looking topology which was introduced by Adji and Raeburn corresponds to the hull-kernel topology in the primitive ideal space Prim T (Γ) of Toeplitz algebra T (Γ) of totally ordered abelian group Γ. In this paper we discuss the closed sets in Prim T (Γ) with the upwards-looking topology and characterize maximal primitive ideals.

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In this paper, we introduce the notion of essence in BE-algebras and related properties are investigated. Also, we discuss relations among subalgebras, filters and essences. Finally, we consider the homomorphic image and inverse image of an essence.

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In this paper, the notion of vague subalgebra of a d-algebra is introduced and some properties are investigated. Using subsets with some conditions, vague subalgebras are constructed and also using a given vague subalgebra, a new vague subalgebra is established.

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In this paper we introduce the concept of free ordered semigroups as follows: If (X,≤X) is an ordered set, an ordered semigroup (F, .,≤F ) is said to be a free ordered semigroup over (X,≤X), if there is an isotone mapping ε : (X,≤X) → (F,≤F ) satisfying the following ”universal” condition: for any ordered semigroup (S, ∗,≤S) and any isotone mapping f : (X,≤X) → (S,≤S), there exists a unique homomorphism ϕ : (F, .,≤F ) → (S, ∗,≤S) such that ϕ◦ε = f. Basing on the fact that the mapping ε is reverse isotone, we find relationships between the mappings ε1 and ε2 which correspond to free ordered semigroups ((F1, .,≤1), ε1) and ((F2, .,≤2), ε2).

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Urban district heating and cooling (DHC) systems operate large freezers, heat exchangers, and boilers to supply hot and cold water, steam etc. stably and economically, based on customers demand. We formulate an operation-planning problem as a nonlinear integer programming problem for an actual DHC plant. To reflect actual decision making appropriately, we incorporate contract-violation penalties into the running cost consisting of fuel and arrangements expenses. Since a yearly operation plan is necessary for checking whether the minimum gas consumption contract is fulfilled or not, we need to solve long-term operation-planning problems. To solve long-term operation-planning problems fast and approximately , we propose a decomposition approach using coarse (monthly) approximate operation-planning problems.

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