Ungar Games in Lattice Theory

• Ungar games are impartial normal-play games on finite lattices and meet-semilattices, where a move sends an element to the meet of itself with a nonempty subset of its covered elements.
• They bridge combinatorial game theory and lattice theory by applying cover geometry to analyze structures such as weak order, Young’s lattice, and Tamari lattices.
• Their study uncovers multiplicative structures, precise enumeration results, and complexity challenges, with open questions relating to PSPACE-completeness and pattern descent conjectures.

Ungar games are impartial normal-play games on finite lattices, and more generally on finite meet-semilattices, in which a move from an element $x$ sends $x$ to the meet of $\{x\}\cup T$ for some nonempty subset $T$ of the elements covered by $x$. The game starts at the top element $\hat 1$ when a finite lattice is given; more generally, an element $x$ of a meet-semilattice with minimum $\hat 0$ is studied via the interval $[\hat 0,x]$. Two players, Atniss and Eeta, alternate making nontrivial Ungar moves, and the first player who cannot move loses. Introduced as a game-theoretic abstraction of Peter Ungar’s move on permutations in weak order, the theory has developed into a lattice-theoretic branch of combinatorial game theory with structural, enumerative, and complexity-theoretic results across several families of posets and lattices (Defant et al., 2023, Choi et al., 2024, Paltrowitz, 3 Sep 2025).

1. Formal definition and basic recursion

For a finite lattice $L$, write $x$0 for the set of elements covered by $x$1. If $x$2, an Ungar move sends $x$3 to

$x$4

Because each $x$5 satisfies $x$6, this is equivalently the meet of the chosen covered elements. The move is trivial if $x$7; only nontrivial moves are legal. The paper writes $x$8 for the set of all positions reachable from $x$9 by an Ungar move. A move is maximal if $\{x\}\cup T$0 (Defant et al., 2023).

The normal-play recursion is the standard one for impartial games, but it is expressed in lattice language. The minimum element $\{x\}\cup T$1 is an Eeta win. More generally, $\{x\}\cup T$2 is an Atniss win if there exists some $\{x\}\cup T$3 that is an Eeta win, and $\{x\}\cup T$4 is an Eeta win if every element of $\{x\}\cup T$5 is an Atniss win. The corresponding sets are denoted

$\{x\}\cup T$6

A useful corollary stated in the original paper is that, for every $\{x\}\cup T$7, the set $\{x\}\cup T$8 is nonempty. This packages the usual fact that every winning position has a move to a losing position, while every losing position is itself already in $\{x\}\cup T$9 (Defant et al., 2023).

This formalism places Ungar games within the ordinary $T$0-theory of impartial normal play, but the move operator is defined by lattice covers and meet, not by a subtraction set, heap split, or octal rule. A plausible implication is that the central difficulty is not recursive game evaluation as such, but the extraction of structural information from the cover geometry of the ambient lattice.

2. Distributive-lattice interpretation and product structure

Ungar games simplify substantially on distributive lattices $T$1 of order ideals of a poset $T$2. If $T$3, then

$T$4

so an Ungar move simply removes some subset of the maximal elements of $T$5: $T$6 On Young diagrams, this means deleting any nonempty subset of exposed corners. This concrete interpretation is the basis of the “Nibble” viewpoint in Young’s lattice and of later binary-string models for shifted staircases (Defant et al., 2023, Choi et al., 2024).

A second structural fact is multiplicative: if $T$7 are lattices, then

$T$8

Equivalently, $T$9 is an Eeta win in the product if and only if each $x$0 is an Eeta win in $x$1. The reason is that

$x$2

and a nontrivial move in the product is a coordinatewise choice of Ungar moves, with at least one coordinate moved nontrivially. This factorization is one of the main tools behind the enumerative results in weak order, Young’s lattice, and Tamari lattices, and it reappears later in the graded-poset classification (Defant et al., 2023).

In distributive lattices one also has that meet is set intersection. Later work on shifted staircases exploits this directly: simultaneous cover deletions indexed by a set $x$3 become a single meet operation, so Ungar moves can be translated into explicit word transformations rather than handled abstractly (Choi et al., 2024).

3. Weak order, Young’s lattice, and Tamari lattices

The first systematic study of Ungar games treated three major lattice families. In weak order on $x$4, Ungar moves are exactly the operations Ungar used originally: one selects some disjoint consecutive decreasing subsequences and reverses them. The principal asymptotic result is

$x$5

The proof proceeds through consecutive pattern avoidance. Defining

$x$6

every permutation that consecutively contains one of the patterns in $x$7 is an Atniss win, so every Eeta win must consecutively avoid every pattern in $x$8, in particular $x$9 (Defant et al., 2023).

In Young’s lattice, an interval $\hat 1$0 is naturally isomorphic to $\hat 1$1, so the Ungar game on such an interval is exactly the Nibble game on a skew Young diagram: one may remove any nonempty subset of exposed corners. The main theorem identifies a broad class of Eeta wins by a path criterion. If $\hat 1$2 is the smallest integer such that $\hat 1$3, and if $\hat 1$4, then $\hat 1$5 is an Eeta win if and only if $\hat 1$6 does not contain an odd-length block of east steps immediately followed by an odd-length block of north steps. This yields exact generating functions for order ideals in rectangles and a corresponding algebraic generating function for type-$\hat 1$7 root posets (Defant et al., 2023).

The rectangular case is summarized by

$\hat 1$8

For type-$\hat 1$9 root posets, the paper derives an algebraic generating function and the asymptotic statement that the fraction of Eeta wins decays exponentially (Defant et al., 2023).

In Tamari lattices $x$0, the key decomposition uses direct sums. If

$x$1

then

$x$2

and

$x$3

The classification of indecomposable Eeta wins is given by the notion of “even-districted.” An indecomposable permutation is an Eeta win in $x$4 if and only if it is even-districted. Enumeration then follows via generating functions: if

$x$5

then $x$6 is algebraic of degree $x$7, satisfying

$x$8

where

$x$9

Moreover,

$\hat 0$0

with $\hat 0$1 and $\hat 0$2, so again the proportion of Eeta wins decays exponentially (Defant et al., 2023).

4. Young-Fibonacci and shifted-staircase classifications

Two conjectural families from the original paper were later resolved completely. The first is the Young-Fibonacci lattice $\hat 0$3, whose elements are words over $\hat 0$4 with rank

$\hat 0$5

For $\hat 0$6, an element $\hat 0$7 is an Eeta win if and only if either $\hat 0$8 and the number of $\hat 0$9's in $[\hat 0,x]$0 is even, or $[\hat 0,x]$1 and the number of $[\hat 0,x]$2's to the left of the leftmost $[\hat 0,x]$3 in $[\hat 0,x]$4 is odd. The corresponding enumeration is

$[\hat 0,x]$5

The proof exploits a sharp dichotomy in the first letter: if $[\hat 0,x]$6, then $[\hat 0,x]$7; if $[\hat 0,x]$8, then $[\hat 0,x]$9 always has enough useful options to be an Atniss win (Choi et al., 2024).

The second family is $L$0, the lattice of order ideals of the shifted staircase

$L$1

A natural bijection identifies order ideals with binary strings $L$2. If $L$3 denotes the set of cover positions and $L$4 the simultaneous local edits defined in the paper, then an order ideal $L$5 with binary representation $L$6 is an Eeta win if and only if

$L$7

and there are no odd-length sequences of $L$8's followed by an odd-length sequence of $L$9's in $x$00. The paper reformulates this via “good” and “bad” strings: $x$01 is an Eeta win exactly when the string ends in $x$02 and its prefix is good. It also records that the sequence $x$03 is OEIS A061279 (Choi et al., 2024).

These classifications are significant because they show that, in highly structured lattices, the Ungar move can collapse to a parity criterion on words or binary strings. This suggests that the meet-of-covers definition, although globally flexible, often admits unexpectedly rigid local descriptions.

5. Graded posets, skeletons, and NAND formulas

A later generalization gives a complete classification of second-player wins on finite graded posets. In the distributive-lattice setting $x$04, the paper identifies a game position with the underlying finite poset itself and defines, for each maximal element $x$05, the maximal subposet $x$06: the subposet of all elements that are $x$07 but $x$08 no other maximal element. It then defines the skeleton $x$09 as the subposet of all elements that are not less than two incomparable elements of $x$10, and the components

$x$11

These notions yield the main recursive theorem: $x$12 A second theorem says that only the skeleton matters: $x$13 A further theorem classifies rooted tree posets by NAND evaluation: if $x$14 is a finite rooted tree poset, then

$x$15

Combining these, one obtains

$x$16

Thus every graded-poset position reduces to rooted-tree components and a conjunction of NAND evaluations (Paltrowitz, 3 Sep 2025).

This framework also extends the two-dimensional Young-diagram theory to all $x$17. For $x$18,

$x$19

The underlying product-structure lemma shows that, in $x$20 with each $x$21 graded, every $x$22 is a disjoint union of posets isomorphic to convex subposets of some $x$23. For $x$24, these convex subposets are chains, so the problem reduces to chain parity (Paltrowitz, 3 Sep 2025).

A related theorem formalizes the slogan that only the skeleton matters: if $x$25 and each element of $x$26 is less than two incomparable elements of $x$27, then

$x$28

In the graded setting, the game can therefore be reduced before evaluation, rather than analyzed on the full poset (Paltrowitz, 3 Sep 2025).

6. Complexity, open problems, and scope of the theory

The original paper proves that Ungar games are $x$29-hard. More precisely, the boolean formula value problem reduces with linear blowup to deciding whether a lattice is an Eeta win. The construction uses small lattices implementing $x$30 and $x$31, with the one-element lattice encoding truth value $x$32 and the two-element lattice encoding $x$33. It then asks whether Ungar games on distributive lattices $x$34 are PSPACE-complete as a function of $x$35, explicitly leaving this as an open question (Defant et al., 2023).

Several structural questions also remain open. In weak order, the upper bound

$x$36

is described as only an upper bound, and the paper asks whether the number of Eeta wins behaves more like $x$37 or like $x$38 for some $x$39. It also suggests improving the estimate by counting permutations avoiding all patterns in

$x$40

rather than only $x$41. A conjecture asserting that Eeta wins have at most $x$42 descents was disproved by Evan Bailey, with the first counterexamples in $x$43 (Defant et al., 2023).

For Young’s lattice, the main path criterion applies to intervals $x$44 with $x$45 sufficiently deep relative to $x$46, but not to arbitrary intervals; extending the criterion to all intervals remains open. The paper also suggests studying $x$47, where no analogous simple path model was available there (Defant et al., 2023). Later work settled two related conjectures, giving complete classifications for the Young-Fibonacci lattice and the shifted staircase lattices $x$48 (Choi et al., 2024).

The graded-poset classification changes the complexity landscape in an important way. It proves that Ungar games on graded posets can be solved in logarithmic space. A plausible implication is that the earlier PSPACE-completeness conjecture cannot hold in the graded setting unless LOGSPACE $x$49 PSPACE. Thus the general theory now has a clear division: broad families of lattices admit crisp combinatorial or logical descriptions of Eeta wins, while the full complexity of the winner problem on arbitrary distributive lattices remains unresolved (Paltrowitz, 3 Sep 2025).