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We are dealing with extensions of commutative rings R ⊆ S whose chains of the poset [ R , S ] of their subextensions are finite (i.e. R ⊆ S has the FCP property) and such that [ R , S ] is a distributive lattice, that we call distributive FCP extensions. Note that the lattice [ R , S ] of a distributive FCP extension is finite. This paper is the continuation of our earlier papers where we studied catenarian and Boolean extensions. Actually, for an FCP extension, the following implications hold: Boolean ⇒ distributive ⇒ catenarian. A comprehensive characterization of distributive FCP extensions actually remains a challenge, essentially because the same problem for field extensions is not completely solved. Nevertheless, we are able to exhibit a lot of positive results for some classes of extensions. A main result is that an FCP extension R ⊆ S is distributive if and only if R ⊆ R ¯ is distributive, where R ¯ is the integral closure of R in S. A special attention is paid to distributive field extensions. [ABSTRACT FROM AUTHOR]

We show that there are finite distributive lattices that are not the congruence lattice of any finite semidistributive lattice. For \(0 \leq k \leq 2\), the distributive lattice \((\mathbf{B}_k)_{++} = \mathbf{2} + \mathbf{B}_k\), where \(\mathbf{B}_k\) denotes the boolean lattice with \(k\) atoms, is not the congruence lattice of any finite semidistributive lattice. Neither can these lattices be a filter in the congruence lattice of a finite semidistributive lattice. However, each \((\mathbf{B}_k)_{++}\) with \(k \geq 3\) is the congruence lattice of a finite semidistributive lattice, say \(\mathbf{L}_k\). These lattices \(\mathbf{L}_k\) cannot be bounded (in the sense of McKenzie), as no \((\mathbf{B}_k)_{++}\) \((k \geq 0)\) is the congruence lattice of a finite bounded lattice. A companion paper shows that every \((\mathbf{B}_k)_{++}\) \((k \geq 0)\) can be represented as the congruence lattice of an infinite semidistributive lattice. We also find sufficient conditions for a finite distributive lattice to be representable as the congruence lattice of a finite bounded (and hence semidistributive) lattice.

Filters play an important role in studying fuzzy logics. From a logical point of view, filters correspond to sets of provable formulae. In this paper, the concept of picture fuzzy sets is applied to residuated lattices. The notion of picture fuzzy filters of a residuated lattice is introduced and some related properties are investigated. The characterizations of picture fuzzy filters are obtained. We show that the set of all the picture fuzzy filters of a residuated lattice forms a distributive lattice. Finally, a unified approach to picture fuzzy filter is proposed and the simple general principles of picture fuzzy filters are given. The relationships between various picture fuzzy filters are studied, by means of the quotient they determine. [ABSTRACT FROM AUTHOR]

Given a finite subset A of a distributive lattice, its total orderization to(A) is a natural transformation of A into a totally ordered set. Recently, the author showed that multivariate maps on distributive lattices which remain invariant under total orderizations generalize various maps on vector lattices, including bounded orthosymmetric multilinear maps and finite sums of bounded orthogonally additive polynomials. Therefore, a study of total orderization invariant maps on distributive lattices provides new perspectives for maps widely researched in vector lattice theory. However, the unwieldy notation of total orderizations can make calculations extremely long and difficult. In this paper we resolve this complication by providing considerably simpler characterizations of total orderization maps. Utilizing these easier representations, we then prove that a lattice multi-homomorphism on a distributive lattice is total orderization invariant if and only if it is symmetric, and we show that the diagonal of a symmetric lattice multi-homomorphism is a lattice homomorphism, extending known results for orthosymmetric vector lattice homomorphisms. [ABSTRACT FROM AUTHOR]

The subgroup lattice of a group is a great source of information about the structure of the group itself. The aim of this paper is to use a similar tool for studying profinite groups. In more detail, we study the lattices of closed or open subgroups of a profinite group and its relation with the whole group. We show, for example, that procyclic groups are the only profinite groups with a distributive lattice of closed or open subgroups, and we give a sharp characterization of profinite groups whose lattice of closed (or open) subgroups satisfies the Dedekind modular law; we actually give a precise description of the behavior of modular elements of the lattice of closed subgroups. We also deal with the problem of carrying some structural information from a profinite group to another one having an isomorphic lattice of closed (or open) subgroups. Some interesting consequences and related results concerning decomposability and the number of profinite groups with a given lattice of closed (or open) subgroups are also obtained. [ABSTRACT FROM AUTHOR]

We introduce soft v-ideals, soft s-ideals and soft vs-ideals over a Menger algebra of rank n. Moreover, we investigate several properties of each of the types of soft ideals. Furthermore, we prove that the collection of all soft vs-ideals forms a distributive lattice. [ABSTRACT FROM AUTHOR]

The premise of the soft set was designed to model vague ideas. In this kind of model, where the set of parameters exhibits some degree of order, lattice order theory is quite beneficial. For researchers studying uncertainty, the notion of soft sets, lattices, and fuzzy sets has proven essential. In this study, the postulation of the Lattice ordered Linear Diophantine Multi-Fuzzy Soft Set (LLDMFSS) is laid out. The fundamental objective of this study is the formation of hybrid algebraic structures for LLDMFSS. Particularly, the characteristics of complete lattice, modular lattice, and distributive lattice were inhibited in LLDMFSS, and some pertinent findings were obtained. Subsequently, the modular and distributive LLDMFSS characterization theorem was implemented. The notion of the direct product for two LLDMFSSs was then addressed. [ABSTRACT FROM AUTHOR]

The aim of this article is to develop a Priestley-style duality for the variety of DN-algebras. In order to achieve this, we use the concept of free distributive lattice extension of a DN-algebra. We establish a connection with the Priestley duality for distributive lattices. Finally, we present topological descriptions for the lattice of filters, for the lattice of congruences, and for certain kinds of subalgebras of a DN-algebra. [ABSTRACT FROM AUTHOR]

We present novel results related to isomorphic resonance graphs of 2-connected outerplane bipartite graphs. As the main result, we provide a structure characterization for 2-connected outerplane bipartite graphs with isomorphic resonance graphs. Three additional characterizations are expressed in terms of resonance digraphs, via local structures of inner duals, as well as using distributive lattices on the set of order ideals of posets defined on inner faces of 2-connected outerplane bipartite graphs. [ABSTRACT FROM AUTHOR]

Closure spaces, namely the finite ones, with closed singletons are studied on the level of segmentations — partitions of the space into closed subsets. Segmentations form a lattice and we study spaces for which this lattice is distributive. Studying these spaces may help understanding mathematical background for segmentation of a digital image. A crucial notion is that of connectively irreducible sets which can be defined in any finite closure space. The paper provides several equivalent conditions for segmentational distributivity in terms of triples of closed sets, connected systems of closed sets, property of induced closure operator on down-sets of connectively irreducible sets, and finally by restriction (or disability) of existence of certain sublattices. [ABSTRACT FROM AUTHOR]

A Pervin space is a set equipped with a bounded sublattice of its powerset, while its pointfree version, called Frith frame, consists of a frame equipped with a generating bounded sublattice. It is known that the dual adjunction between topological spaces and frames extends to a dual adjunction between Pervin spaces and Frith frames, and that the latter may be seen as representatives of certain quasi-uniform structures. As such, they have an underlying bitopological structure and inherit a natural notion of completion. In this paper we start by exploring the bitopological nature of Pervin spaces and of Frith frames, proving some categorical equivalences involving zero-dimensional structures. We then provide a conceptual proof of a duality between the categories of T 0 complete Pervin spaces and of complete Frith frames. This enables us to interpret several Stone-type dualities as a restriction of the dual adjunction between Pervin spaces and Frith frames along full subcategory embeddings. Finally, we provide analogues of Banaschewski and Pultr’s characterizations of sober and T D topological spaces in the setting of Pervin spaces and of Frith frames, highlighting the parallelism between the two notions. [ABSTRACT FROM AUTHOR]

A differential operator of weight λ is the algebraic abstraction of the difference quotient dλ(f)(x) := f(x + λ) − f(x)/λ, including both the derivation as λ approaches to 0 and the difference operator when λ = 1. Correspondingly, differential algebra of weight λ extends the well-established theories of differential algebra and difference algebra. In this paper, we initiate the study of differential operators with weights, in particular difference operators, on lattices. We show that differential operators of weight − 1 on a lattice coincide with differential operators, while differential operators are special cases of difference operators. Distributivity of a lattice is characterized by the existence of certain difference operators. Furthermore, we characterize and enumerate difference operators on finite chains and finite quasi-antichains. [ABSTRACT FROM AUTHOR]

In order to find an analogue of the structure theorem,"a semigroup is a full subdirect product of a semilattice and a group if and only if it is an E -inversive sturdy semilattice of cancellative monoids (Theorem 14 of [H. Mitsch, Subdirect products of E-inversive semigroups, J. Austral. Math. Soc. 48 (1990) 66–78])", in the setting of seminearrings we have characterized the seminearrings which are full subdirect products of a bi-semilattice and a (zero-symmetric) near-ring and which are subdirect products of a distributive lattice and a (zero-symmetric) near-ring. [ABSTRACT FROM AUTHOR]

The theory of rough sets has been classified into several directions where the information granules are generated using tolerances, pre-orders and binary relations. Maximal Compatibility Block (MCB) plays an important role in case of tolerance relations. In the present work, the lattice structure of rough sets generated by MCBs is explored. The concepts are applied to establish equivalence between the isomorphism of graphs with isomorphism of rough lattices constructed using MCBs. [ABSTRACT FROM AUTHOR]

Given any finite subset A of order n of a distributive lattice and k ∈ { 1 , … , n } , there is a natural extension of the median operation to n variables which generalizes the notion of the kth smallest element of A. By applying each of these operations to A, a totally ordered set to(A) is obtained. We refer to to(A) as the total orderization of A. After developing a brief theory of total orderization invariant maps on distributive lattices, it is shown in this paper how these functions generalize and provide new characterizations for symmetric continuous positively homogeneous functions, bounded orthosymmetric multilinear maps, and certain power sum polynomials on vector lattices. These theorems generalize several results by Bernau, Huijsmans, Kusraev, Azouzi, Boulabiar, Buskes, Boyd, Ryan, and Snigireva and in turn reveal novel properties of the various maps studied in this paper. [ABSTRACT FROM AUTHOR]

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Associated with any composition β = (a , b , ...) is a corresponding fence poset F (β) whose covering relations are x 1 ⊲ x 2 ⊲ ... ⊲ x a + 1 ⊳ x a + 2 ⊳ ... ⊳ x a + b + 1 ⊲ x a + b + 2 ⊲ .... The distributive lattice L (β) of all lower order ideals of F (β) is important in the theory of cluster algebras. In addition, its rank generating function r (q ; β) is used to define q-analogues of rational numbers. Kantarcı Oğuz and Ravichandran recently showed that its coefficients satisfy an interlacing condition, proving a conjecture of McConville, Smyth, and Sagan, which in turn implies a previous conjecture of Morier-Genoud and Ovsienko that r (q ; β) is unimodal. We show that, when β has an odd number of parts, then the polynomial is also partially symmetric: the number of ideals of F (β) of size k equals the number of filters of size k, when k is below a certain value. Our proof is completely bijective. Kantarcı Oğuz and Ravichandran also introduced a circular version of fences and proved, using algebraic techniques, that the distributive lattice for such a poset is rank symmetric. We give a bijective proof of this result, as well. We end with some questions and conjectures raised by this work. [ABSTRACT FROM AUTHOR]

We completely classify all varieties of aperiodic monoids with central idempotents whose subvariety lattice is distributive. [ABSTRACT FROM AUTHOR]

We consider profunctors between posets and introduce their graph and ascent. The profunctors Pro (P , Q) form themselves a poset, and we consider a partition I ⊔ F of this into a down-set I and up-set F , called a cut. To elements of F we associate their graphs, and to elements of I we associate their ascents. Our basic results is that this, suitably refined, preserves being a cut: We get a cut in the Boolean lattice of subsets of the underlying set of Q × P . Cuts in finite Booleans lattices correspond precisely to finite simplicial complexes. We apply this in commutative algebra where these give classes of Alexander dual square-free monomial ideals giving the full and natural generalized setting of isotonian ideals and letterplace ideals for posets. We study Pro (N , N) . Such profunctors identify as order preserving maps f : N → N ∪ { ∞ } . For our applications when P and Q are infinite, we also introduce a topology on Pro (P , Q) , in particular on profunctors Pro (N , N) . [ABSTRACT FROM AUTHOR]

In this article, the concept of weak annulet is defined on the distributive lattice with dual weak complementation (DDWCL). Properties of weak annulets are proved. The relationship between orthopseudoring and ortho-lattice of all weak annulets of DDWCL is demonstrated. Congruence relations, with respect to weak annulets on DDWCL (W-congruences), are established. The double-face algebraic structure of all weak annulets and all W-congruence is investigated.

We study the lattice structure of fuzzy A-ideals in an mv-module M (fai (M), symbolically) and show that it is a complete Heyting lattice and so the set of its pseudocomplements forms a Boolean algebra. In the sequel, the properties of fuzzy congruences in an mv-module are investigated and using them some structural theorems are stated and proved. Finally, it is proved that fai (M) can be embedded into the lattice of fuzzy congruences. [ABSTRACT FROM AUTHOR]

The structural theory of semirings with additional conditions is considered. Multiplicatively idempotent semirings with the annihilator condition are studied. General properties of such semirings are considered. Examples are given. A criterion for the fulfillment of the annihilator condition in an arbitrary multiplicatively idempotent semiring with zero is proved (Proposition 6). In terms of annihilators, new abstract characterizations of semirings isomorphic to the direct product of a Boolean ring with unit and a Boolean lattice (Theorem 1) are obtained. The direct product of a Boolean ring and a distributive lattice with the annihilator condition is a multiplicatively idempotent semiring with the annihilator condition. Generally speaking, the opposite is not true (Theorem 2). An example of a general nature of a multiplicatively idempotent semiring with unit and with the annihilator condition, which is not isomorphic to the direct product of a Boolean ring and a distributive lattice, is constructed. In conclusion, a number of supplements are given. [ABSTRACT FROM AUTHOR]

The Vietoris space of a Stone space plays an important role in the coalgebraic approach to modal logic. When generalizing this to positive modal logic, there is a variety of relevant hyperspace constructions based on various topologies on a Priestley space and mechanisms to topologize the hyperspace of closed sets. A number of authors considered hyperspaces of Priestley spaces and their application to the coalgebraic approach to positive modal logic. A mixture of techniques from category theory, pointfree topology, and Priestley duality have been employed. Our aim is to provide a unifying approach to this area of research relying only on a basic familiarity with Priestley duality and related free constructions of distributive lattices. [ABSTRACT FROM AUTHOR]

The notion of a pre-period of a finite bounded distributive lattice (BDL) A is defined by means of the notion of a pre-period of a finite connected monounary algebra: it is the maximum value of the pre-period of an endomorphism and 0-fixing connected mapping of A to A. The main result is that the pre-period of any finite BDL is less than or equal to the length of the lattice; also, necessary and sufficient conditions under which it is equal to the length of the lattice, are shown. [ABSTRACT FROM AUTHOR]

We give an implemented algorithm for computing the distributive lattice associated to a Cohen-Macaulay bipartite graph and, consequently, the set of all minimal vertex covers of a Cohen-Macaulay bipartite graph [ABSTRACT FROM AUTHOR]

In this paper, we study the concept of O-fuzzy ideals in distributive lattices with the least element 0 as a generalization of δ-fuzzy ideals. We investigate the properties of O-fuzzy ideals and establishes their relationships with other types of fuzzy ideals. We observe that $ \chi_{\{0\}} $ χ { 0 } may not be an O-fuzzy ideal. Hence, we derive a necessary and sufficient condition for $ \chi_{\{0\}} $ χ { 0 } to become an O-fuzzy ideal. It is proved that the class of O-fuzzy is a distributive lattice. Moreover, we establish a necessary and sufficient condition for the class of O-fuzzy ideals to be a complete distributive lattice. [ABSTRACT FROM AUTHOR]

Numerous scholars have studied fuzzy congruence relations in various algebraic structures. In this paper, we introduce a new kind of fuzzy congruence relations on a fuzzy algebraic structure (i.e. fuzzy lattice). Additionally, we provide a characterization of fuzzy congruence relations using its level sets and support. We also make a theoretical study of their basic properties analogous to those of ordinary congruence relations. Furthermore, we study the lattice of fuzzy congruence relations and also, we show that the lattice of all fuzzy congruence relations is complete, distributive and a special class of this lattice forms a modular lattice under certain conditions. [ABSTRACT FROM AUTHOR]

Recently, De Miguel, Bustince and De Baets have conducted a systematic study on convolution lattices based on distributive lattices. There have been few reports on applying non-distributive lattices to a domain of functions. As a complement to their work, in this paper, we carry out an in-depth investigation of convolution operations of the functions between a non-distributive lattice (domain) and a frame (co-domain). We first present an equivalence characterization between non-distributive lattices and idempotent functions and further show that a subset of the set of idempotent functions is closed under convolution operations. We demonstrate that this subset also is a bisemilattice and satisfies the Birkhoff equation under join- and meet-convolution operations. Finally, we analyze and study the lattice structure related to the obtained algebraic structure. [ABSTRACT FROM AUTHOR]

This study aims to examine the concept of O-fuzzy ideals of " L B -valued general fuzzy automata" (for simplicity, L B -valued GFA), where B is regarded as a set of propositions about the GFA, and its underlying structure is a complete distributive lattice as a generalization of δ -fuzzy ideals. We provide a necessary and sufficient condition for χ { 0 } to be an O-fuzzy ideal, and we investigate the relationship between the concepts of O-fuzzy ideals and α -fuzzy ideals in a set B. Accordingly, the related notions and the results obtained in this study are clarified and explicated by some examples. [ABSTRACT FROM AUTHOR]

A chief factor of a group is said to be -central if . In 1997, Shemetkov posed the problem of describing finite group formations such that coincides with the class of groups for which all chief factors are -central. We refer to such formations as centrally saturated. We prove that the centrally saturated formations form a complete distributive lattice. As an answer to a question posed by Ballester-Bolinches and Perez-Ramos, conditions for a centrally saturated formation to be saturated and solvably saturated in the class of all groups are found. As a consequence, a criterion for hereditary Fitting formations to be solvably saturated is obtained. [ABSTRACT FROM AUTHOR]

In this paper we present a construction for partial order in T[X, Y] which is called the ternary semigroup of mappings, derived from the partial orders defined in X and Y. Mainly we study various lattice structures in T[X, Y]. The last section deals with the lattice isomorphism problems and the notion of partially ordered ternary semigroup in T[X, Y]. [ABSTRACT FROM AUTHOR]

A discrete Sugeno integral on a bounded distributive lattice L is defined as an idempotent weighted lattice polynomial. Another possibility for axiomatization of Sugeno integrals is to consider compatible aggregation functions, uniquely extending the L-valued fuzzy measures. This paper aims to study the mentioned unique extension property concerning the possible extension of a Sugeno integral to non-distributive lattices. We show that this property is equivalent to the distributivity of the underlying bounded lattice. As a byproduct, an alternative proof of Iseki's result, stating that a lattice having prime ideal separation property for every pair of distinct elements is distributive, is provided. [ABSTRACT FROM AUTHOR]

The theory of bounded, distributive lattices provides the appropriate language for describing directionality and asymptotics in dynamical systems. For bounded, distributive lattices the general notion of 'set-difference' taking values in a semilattice is introduced, and is called the Conley form. The Conley form is used to build concrete, set-theoretic models of spectral spaces, or Priestley spaces, of bounded, distributive lattices and their finite coarsenings. Such representations formulate and compute order-theoretic models of dynamical systems such as Morse decompositions and Morse representations, which may be regarded as global characteristics of a dynamical system. [ABSTRACT FROM AUTHOR]

In this article, we introduce a lattice congruence θ I d with respect to a nonempty ideal I of a distributive lattice L and a derivation d on L. We investigate some necessary and sufficient conditions for the quotient algebra L / θ I d to be a Boolean algebra. [ABSTRACT FROM AUTHOR]

It is known that every non-universal self-full degree in the structure of the degrees of computably enumerable equivalence relations (ceers) under computable reducibility has exactly one strong minimal cover. This leaves little room for embedding wide partial orders as initial segments using self-full degrees. We show that considerably more can be done by staying entirely inside the collection of non-self-full degrees. We show that the poset can be embedded as an initial segment of the degrees of ceers with infinitely many classes. A further refinement of the proof shows that one can also embed the free distributive lattice generated by the lower semilattice as an initial segment of the degrees of ceers with infinitely many classes. [ABSTRACT FROM AUTHOR]

Ordered by set inclusion, the retracts of a lattice L together with the empty set form a bounded poset R e t (L) . By a grid we mean the direct product of two non-singleton finite chains. We prove that if G is a grid, then R e t (G) is a lattice. We determine the number of elements of R e t (G) . Some easy properties of retracts, retractions, and their kernels called retraction congruences of (mainly distributive) lattices are found. Also, we present several examples, including a 12-element modular lattice M such that R e t (M) is not a lattice. [ABSTRACT FROM AUTHOR]

In this paper we study prime spectra of commutative BCK-algebras. We give a new construction for commutative BCK-algebras using rooted trees, and determine both the ideal lattice and prime ideal lattice of such algebras. We prove that the spectrum of any commutative BCK-algebra is a locally compact generalized spectral space which is compact if and only if the algebra is finitely generated as an ideal. Further, we show that if a commutative BCK-algebra is involutory, then its spectrum is a Priestley space. Finally, we consider the functorial properties of the spectrum and define a functor from the category of commutative BCK-algebras to the category of distributive lattices with zero. We give a partial answer to the question: what distributive lattices lie in the image of this functor? [ABSTRACT FROM AUTHOR]

Dyrda and Prucnal gave a Hilbert-style axiomatization for the { ∧ , ∨ } -fragment of classical propositional logic. Their proof of completeness follows a different approach to the standard one proving the completeness of classical propositional logic. In this note, we present an alternative proof of Dyrda and Prucnal's result following the standard arguments which prove the completeness of classical propositional logic. [ABSTRACT FROM AUTHOR]

We establish that the isomorphism problem for the classes of computable nilpotent rings, distributive lattices, nilpotent groups, and nilpotent semigroups is Σ 1 1 -complete, which is as complicated as possible. The method we use is based on uniform effective interpretations of computable binary relations into computable structures from the corresponding algebraic classes. [ABSTRACT FROM AUTHOR]

Given a partially order set (poset)P, and a pair of families of ideals I and filters F in P such that each pair (I,F)∈I×F has a non-empty intersection, the dualization problem over P is to check whether there is an ideal X in P which intersects every member of F and does not contain any member of I. Equivalently, the problem is to check for a distributive lattice L=L(P), given by the poset P of its set of joint-irreducibles, and two given antichains A,B⊆L such that no a∈A is dominated by any b∈B, whether A and B cover (by domination) the entire lattice. We show that the problem can be solved in quasi-polynomial time in the sizes of P, A and B, thus answering an open question in Babin and Kuznetsov (2017). As an application, we show that minimal infrequent closed sets of attributes in a rational database, with respect to a given implication base of maximum premise size of one, can be enumerated in incremental quasi-polynomial time. [ABSTRACT FROM AUTHOR]

In computability theory, the standard tool to classify preorders is provided by the computable reducibility. If R and S are preorders with domain ω , then R is computably reducible to S if and only if there is a computable function f such that for all x and y , x R y ⇔ f (x) S f (y). We study the complexity of preorders which arise in a natural way in computable structure theory. We prove that the relation of computable isomorphic embeddability among computable torsion abelian groups is a Σ 3 0 complete preorder. A similar result is obtained for computable distributive lattices. We show that the relation of primitive recursive embeddability among punctual structures (in the setting of Kalimullin et al.) is a Σ 2 0 complete preorder. [ABSTRACT FROM AUTHOR]

Distributive lattices form an important, well-behaved class of lattices. They are instances of two larger classes of lattices: congruence-uniform and semidistributive lattices. Congruence-uniform lattices allow for a remarkable second order of their elements: the core label order; semidistributive lattices naturally possess an associated flag simplicial complex: the canonical join complex. In this article we present a characterization of finite distributive lattices in terms of the core label order and the canonical join complex, and we show that the core label order of a finite distributive lattice is always a meet-semilattice.

Since the seminal work of Gale and Shapley, stable assignments have received widespread attention for their mathematical elegance and broad applicability. However, in applications such as the school choice problem, in which public schools are often perceived as commodities and only students' welfare matters, enforcing stability implies a loss of efficiency for the students. In "Legal assignments and fast EADAM with consent via classical theory of stable matchings," Faenza and Zhang study two extensions of the traditional model—legal assignments and efficiency adjusted deferred acceptance mechanism (EADAM)—that strive to regain this loss in efficiency. The authors establish a tight connection between legal and stable assignments, which allows them to use critical structural tools of stable matchings, such as the concept of rotations, to design provably fast algorithms for (1) optimizing linear functions over the set of legal assignments and (2) finding the outcome of EADAM. These algorithmic results greatly improve the applicability of both extensions as witnessed by a complexity analysis and experimental results. Gale and Shapley's stable assignment problem has been extensively studied, applied, and extended. In the context of school choice, mechanisms often aim at finding an assignment that is more favorable to students. We investigate two extensions introduced in this framework—legal assignments and the efficiency adjusted deferred acceptance mechanism (EADAM) algorithm—through the lens of the classic theory of stable matchings. In any instance, the set L of legal assignments is known to contain all stable assignments. We prove that L is exactly the set of stable assignments in another instance. Moreover, we show that essentially all optimization problems over L can be solved within the same time bound needed for solving it over the set of stable assignments. A key tool for this latter result is an algorithm that finds the student-optimal legal assignment. We then generalize our algorithm to obtain the assignment output of EADAM with any given set of consenting students without sacrificing the running time, hence largely improving in both theory and practice over known algorithms. Finally, we show that the set L can be much larger than the set of stable matchings, connecting legal matchings with certain concepts and open problems in the literature. [ABSTRACT FROM AUTHOR]