In this study, we fabricated a superhydrophilic and underwater superoleophobic protonated melamine sponge for effective separation of water-rich immiscible oil/water mixtures with extremely high separation efficiency. This protonated melamine sponge exhibited excellent antifouling properties and could be used to separate oil/water mixtures continuously for up to 12 h without any increase in the oil content in filtrate. Moreover, our compressed protonated melamine sponge could separate both surfactant-free and -stabilized oil-in-water emulsions with high separation efficiencies. The high performance of this protonated melamine sponge and its efficient, energy- and cost-effective preparation suggest that it has great potential for use in practical applications.

Functional materials with a superwetting surface property have been extensively explored to achieve emulsion separation. In this paper, we report a simple and inexpensive method for fabricating superhydrophobic/superoleophilic porous materials from polymeric sponges. These microstructured porous materials, which do not contain any fluorinated compounds, maintain their superhydrophobicity and superoleophilicity after long-term organic solvent immersion and display environmental stability. These superhydrophobic porous materials can effectively separate a wide range of water-in-oil emulsions including surfactant-free and surfactant-stabilized water-in-oil emulsions with high efficiency (>99.98%) and high flux (up to 155 000 L m h bar). Meanwhile, these materials exhibited excellent pH resistance and antifouling properties. The high performance of our superhydrophobic porous materials and their efficient, low-energy, cost-effective preparation suggest that they have a great potential for practical applications.

Profinite equations are an indispensable tool for the algebraic classification of formal languages. Reiterman's theorem states that they precisely specify pseudovarieties, i.e. classes of finite algebras closed under finite products, subalgebras and quotients. In this paper Reiterman's theorem is generalised to finite Eilenberg-Moore algebras for a monad T on a variety D of (ordered) algebras: a class of finite T-algebras is a pseudovariety iff it is presentable by profinite (in-)equations. As an application, quasivarieties of finite algebras are shown to be presentable by profinite implications. Other examples include finite ordered algebras, finite categories, finite ∞-monoids, etc.For example, for D = sets, posets and monoids we get D = Stone spaces, Priestley spaces and profinite monoids. Next, we consider a monad T on D and associate to it a monad T on D, called the profinite monad of T. For example, if D = Set and T is the finite word monad (whose algebras are precisely monoids), then T is the monad of profinite monoids on the category of Stone spaces; that is, T associates to each finite Stone space (= finite set) X the space X * of profinite words on X. Similarly, for the monad T of finite and infinite words on Set (whose algebras we call ∞-monoids) the profinite monad T constructs the space of profinite ∞-words.

The Schützenberger product of monoids is a key tool for the algebraic treatment of language concatenation. In this paper we generalize the Schützenberger product to the level of monoids in an algebraic category D, leading to a uniform view of the corresponding constructions for monoids (Schützenberger), ordered monoids (Pin), idempotent semirings (Klíma and Polák) and algebras over a field (Reutenauer). In addition, assuming that D is part of a Stone-type duality, we derive a characterization of the languages recognized by Schützenberger products.Liang-Ting Chen acknowledges support from AFOSR. Henning Urbat acknowledges support from DFG under project AD / -. arXiv:1605.01810v1 [cs.FL] 6 May 2016 Schützenberger Products in a Category theorems for regular languages. See also [ , ] for related duality-based work. Recently, Bojańczyk [ ] proposed to use monads instead of monoids to get a categorical grasp on languages beyond finite words. By combining this idea with our duality framework, we established in [ , ] a variety theorem that covers most Eilenberg-type correspondences known in the literature, e.g. for languages of finite words, infinite words, words on linear orderings, trees, and cost functions. PreliminariesIn this paper we study monoids and language recognition in algebraic categories. The reader is assumed to be familiar with basic universal algebra and category theory; see the Appendix for a toolkit. We call a variety D of algebras or ordered algebras commutative if, for any two algebras A, B ∈ D, the set [A, B] of morphisms from A to B forms an algebra of D with operations taken pointwise in B. Our applications involve the commutative varieties Set (sets), Pos (posets, as ordered algebras without any operation), JSL (join-semilattices with 0), K-Vec (vector spaces over a field K) and S-Mod (modules over a commutative semiring S with 0, 1). Note that JSL and K-Vec are special cases of S-Mod for S = {0, 1}, the two-element semiring with 1 + 1 = 1, and S = K, respectively.Notation . . Let A , B, C , D always denote commutative varieties of algebras or ordered algebras. We write Ψ = Ψ D : Set → D for the left adjoint to the forgetful functor |−| : D → Set; thus Ψ X is the free algebra of D over X. For simplicity, we assume that X is a subset of |Ψ X| and the universal map X |Ψ X| is the inclusion. Denote by 1 D = Ψ 1 the free one-generated algebra. Example . . ( ) For D = Set or Pos we have Ψ X = X (discretely ordered). ( ) For D = JSL we get Ψ X = (P f X, ∪), the semilattice of finite subsets of X. ( ) For D = S-Mod we have Ψ X = S (X) , the S-module of all finite-support functions X → S with sum and scalar product defined pointwise. Definition . . Let A, B, C ∈ D. By a bimorphism from A, B to C is meant a function f : |A| × |B| → |C| such that the maps f (a, −) : |B| → |C| and f (−, b) : |A| → |C| carry morphisms of D for every a ∈ |A| and b ∈ |B|. A tensor product of A and B is a universal bimorphism t A,B : |A| × |B| → |A ⊗ B|, in the sense that for any bimorphism f : |A|×|B| → |C| there is a unique f...

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